Every calculator on this site rests on either a standard formula, a published physical constant, an engineering/regulatory standard, or a specific real-world data point used for calibration. This page lists the primary source for each, grouped by area. Individual tools also link back to the relevant row here from their own footer.
Referenced across multiple areas rather than tied to one tool.
| Constant / standard | Value used | Source |
|---|---|---|
| Cooking a steak is transient heat conduction: time to a target core temperature scales with roughly the square of the thickness, and beef has a thermal diffusivity near 1.2×10-7 m²/s, varying with fat content | Standard transient-conduction treatment (Incropera & DeWitt, Fundamentals of Heat and Mass Transfer) applied to published thermal properties of beef; the page solves the heat equation numerically rather than using the one-term Heisler approximation, which is only valid for Fourier numbers above ~0.2 and so does not cover most of a sear. Doneness temperature bands follow standard culinary practice | Steak Timer |
| The temperature to take the steak out at, and the carryover it will then gain | Not a published carryover figure subtracted from the target. The page simulates the whole rest from every moment the steak could leave the pan and selects the one whose centre finishes on the target, by bracketed search over the simulated frames — the rest is monotonic in the pull point, so the search is exact rather than iterative. The carryover this produces ranges from about −2 °C on an 8 mm steak (thin steaks cool while they rest) to +23 °C on a 40 mm one, against the 3–6 °C usually published, because a 230 °C cast-iron sear leaves roughly a third of the meat past well done and that shell releases its heat inward; both figures are shown side by side and labelled. Evaporation during the rest is modelled as a wicking-limited sink — the lesser of a Chilton-Colburn wet-surface rate and a transport limit through the crust — calibrated so the steak loses the 1–2% of its mass that a resting steak measurably loses (1.34% on a 25 mm steak). Adding it moved the 25 mm carryover by about two degrees, so it does not explain the gap with the published figure; the gap is the cook. steak_model.py reproduces the solve independently (41.4 °C at 398 s against the page’s 41.6 °C at 401 s) and checks that the finish never exceeds the steak’s mean temperature at the pull, which is the most it could reach with no losses at all. Cooking times are guidance, not food-safety advice | Steak Timer |
| The weight a typical steak of each cut has at a given thickness, offered as a check on the thickness | Typical footprint × thickness × the cut’s density (the same table the cooking model uses, 1,030–1,060 kg/m³). The footprints are typical butcher’s portions, not measurements: fillet about 74 mm across (set so a 50 mm fillet, the page’s default, is 8.0 oz), sirloin 135 × 80 mm (set so a 25 mm sirloin is the 250 g the page has always opened on), ribeye 140 × 105, rump 150 × 100, flank 250 × 90, T-bone 170 × 120 mm with about 11% of the outline in bone at an assumed 1,500 kg/m³ (about 15% of the weight). At a fixed size the weight follows the thickness one for one, which is what lets a known weight point at a mis-measured thickness. steak_model.py reproduces the sums and the inverse. A check, not a measurement: real steaks vary in size | Steak Timer |
| UK student loans repay 9% of income above the plan threshold (6% for postgraduate loans, charged concurrently), with any balance cancelled after 25–40 years depending on plan | GOV.UK “Repaying your student loan — what you pay” and the Student Loans Company repayment plan rules; UK thresholds shown are 2026/27 (read 6 September 2026) and are uprated each April, which is why the tool leaves every one of them editable. The Australian, New Zealand and US figures in that tool are older and are labelled with their own year on the page | Student Loan Calculator |
| Rod-side (annulus) area, retract force and speed, and the 2:1 cylinder | Annulus area π(D²−d²)/4 for a single-rod cylinder; force = P·A and speed = Q/A on each face. Standard fluid-power practice (Parker and Eaton design handbooks; ISO 6020/6022 cylinder series). hydraulic_model.py checks that bore = annulus + rod exactly, that rod = bore/√2 gives an area ratio of precisely 2, and — the identity that ties it together — that force × speed equals P×Q in both directions and in regeneration, so the model can never invent power | Hydraulic Force Calculator |
| Regeneration net area equals the rod area; return-line flow exceeds supply on retract | With the rod side tied back to the cap side both faces see the same pressure, so the net area is bore minus annulus — identically the rod area, which hydraulic_model.py asserts to 1e-18. Retract return flow is supply × (bore area / annulus area); sizing the return for the supply flow instead is a common error that builds back-pressure on the annulus face and subtracts directly from the delivered force | Hydraulic Force Calculator |
| Sequence-of-returns risk reverses sign between saving and drawing down | With no cash flows the order of returns cannot matter at all — a product of growth factors is invariant under reordering, which fire_model.py verifies exactly over 40 random returns. Flows are what make order matter, and the direction of the flow sets the sign: contributions into a slump buy in cheap, withdrawals from one sell a larger share of a shrunken pot. Verified on a case checkable by hand — 100 through +50%/−50% contributing 10 a year ends at 90 and reversed at 100; withdrawing 10 instead it is 60 and 50. The underlying drawdown treatment follows Bengen (1994) and the Trinity study | FIRE Calculator |
| Design heat loss = ΣU·A·ΔT + 0.33·n·V·ΔT; emitter output scales with the 1.3 power of the water-to-air temperature difference against its ΔT50 rating | Standard steady-state building heat loss method (CIBSE Guide A / BS EN 12831); 0.33 Wh/m³K is the volumetric heat capacity of air. The radiator exponent of ~1.3 is the conventional value used in EN 442 output correction | Heat Loss & Heat Pump Sizing |
| Only the exposed surfaces count: a party wall between two heated homes carries essentially no heat, and neither does the floor or ceiling between heated flats | A consequence of the same ΔT that drives every other term — the method in BS EN 12831 counts each building element against the temperature on its far side, and adjacent heated space is taken as at or near internal temperature | Heat Loss & Heat Pump Sizing |
| The ground floor is corrected by about a half, and junctions add roughly 12% to the fabric loss | BS EN 12831 applies a reduction factor to ground-coupled elements because the earth stays well above outdoor air through the heating season; the thermal-bridging allowance is a typical whole-dwelling figure for ordinary construction and is considerably higher where junctions are poorly detailed | Heat Loss & Heat Pump Sizing |
| Seasonal efficiency is modelled as about 40% of the Carnot limit between outdoor air and flow temperature, not as a straight line | The Carnot ceiling for a heat pump is T_hot/(T_hot − T_cold) in kelvin; field-trial and manufacturer seasonal figures for air-source units sit near 40% of it. The shape matters: the real curve steepens as flow temperature falls, which a linear fit understates at 35 °C | Heat Loss & Heat Pump Sizing |
| Standard gravity, g | 9.80665 m/s² | NIST CODATA — exact by international definition (CGPM, 1901) |
| Gravitational constant, G | 6.674 × 10⁻¹¹ N·m²/kg² | NIST CODATA |
| Universal gas constant, R | 8.314 J/(mol·K) | NIST CODATA |
| Standard atmosphere model | ICAO Standard Atmosphere | ICAO Doc 7488 — the standard reference for pressure/temperature vs. altitude |
| SI unit conversion factors | exact, by definition | NIST SI units guide |
| Claim | Source | Tool |
|---|---|---|
| Country population, land and surface area, GDP, GDP per capita, life expectancy, urban share, CO₂ per capita and forest cover | World Bank Open Data (CC BY 4.0), each figure taken at the most recent year the Bank reports for that country, with the year shown alongside it | World Atlas |
| Country boundaries at 1:110m | Natural Earth (public domain), via the world-atlas TopoJSON build. 177 shapes: states smaller than the resolution are absent from the map and present in the data | World Atlas |
| Highest and lowest points, ISO 3166-1 codes and continent assignment | Wikidata (CC0), properties P610, P1589, P2044, P298, P299 and P30 | World Atlas |
| Equal Earth projection constants | Šavrič, Patterson & Jenny, The Equal Earth map projection, International Journal of Geographical Information Science 33(3), 2019 — A₁=1.340264, A₂=−0.081106, A₃=0.000893, A₄=0.003796 | World Atlas |
| Membership of the EU, NATO, G7, G20, OECD, OPEC, ASEAN, BRICS, the Commonwealth and the Nordic Council | Each organisation's own published member list; the member counts are asserted against the published totals in models/atlas_model.py, so a stale list fails the build | World Atlas |
| Tracking a storm across successive strikes | Least squares through the logged (time, distance) pairs; the slope is the closing speed. The verdict is withheld unless the fitted movement exceeds the scatter about the line by half again, because consecutive strikes are different discharges in a storm that may be tens of kilometres across — the spread between them is routinely larger than the storm's real motion over a couple of minutes, so a slope through two points measures the storm's width and not its travel. Storm speeds of 20–50 km/h are typical. The 30/30 guidance is public safety advice, quoted rather than derived. Verified in lightning_model.py, including that a stationary storm's scatter is correctly refused. | Lightning Distance |
| Speed of sound in dry air: c = 331.3 √(1 + T/273.15) m/s | The standard ideal-gas result, exact for dry air. Pressure does not enter it at all; humidity raises the speed by under 1% even at saturation, which is well inside the error of counting out loud, so neither is modelled. The linear form everyone quotes (331.3 + 0.606 T) is within 0.1% up to about 25 °C and 0.35% out by 50. | Lightning Distance |
| “Divide by 3” is out by 2%; “divide by 5” is out by 5.4% | Definitional from the speed above at 15 °C: sound covers a kilometre in 2.94 s and a mile in 4.73 s. The Rulebook entry previously offered the two divisors as a matched pair; building this tool showed the mile form is more than twice as far out, and the entry now says so. Both underestimate the distance, which is the safe direction. | Lightning Distance |
| Temperature moves the answer by 9.1% across −10 to 40 °C; one second of miscounting moves it by 11.1% | Both computed from the same speed relation. The comparison is the point of the page: the input a reader is asked for matters less than the input nobody can control, so the tool reports a band rather than a figure. | Lightning Distance |
| The 30/30 guidance: shelter under 30 s, wait 30 minutes after the last thunder | Public safety guidance from national weather services, quoted rather than derived. Nothing on the page calculates it. Thirty seconds works out at 10.2 km at 15 °C, which is inside the distance strikes are recorded reaching from the parent storm. | Lightning Distance |
| Strikes reach ~16 km from the storm; thunder usually stops carrying by ~15 km | Order-of-magnitude figures from the meteorological literature on positive cloud-to-ground discharges (“bolt from the blue”) and on the upward refraction of thunder by warmer air aloft. The asymmetry is the important part and it is the right way round: thunder becomes inaudible before lightning becomes harmless, so a silent storm can still be in range. | Lightning Distance |
| Lunar position (moonrise, moonset, illuminated fraction) | Paul Schlyter, Computing planetary positions — Keplerian elements plus the twelve largest perturbations of lunar longitude (evection, variation, the yearly and parallactic equations among them), five of latitude and two of distance, stated by the author as good to about 2 arcminutes. That is a rise time good to a few minutes and a phase to within 10 minutes. The Sun is taken from the same source. It is not an ephemeris and will not time an occultation or the contacts of an eclipse to the second. | Moon Phase |
| Moonrise altitude = 0.7275 × parallax − 0.5667° | Standard formulation (Meeus). Unlike the Sun, the Moon is close enough that horizontal parallax — nearly a degree — matters, and it works against refraction: the net threshold is about +0.125°, so the Moon is said to rise while its centre is still slightly above the geometric horizon, the opposite of the Sun. | Moon Phase |
| Moonrise and moonset verified against the US Naval Observatory | USNO rise/set/transit service. London, 28 August 2026: USNO gives moonrise 18:57 UT and moonset 05:16 UT; this tool computes both to within two minutes. The awkward case is checked too — USNO lists no moonrise for London on 6 September 2026, and the tool reports none rather than inventing one. | Moon Phase |
| Psychrometric formulas (Magnus/Arden Buck approximation) | Buck, A.L. (1981), J. Applied Meteorology 20(12) — standard vapour-pressure approximation | Psychrometer |
| Wind chill formula | NWS/Environment Canada 2001 joint formula | Feels Like Temperature |
| Heat index formula | Rothfusz, L.P. (1990), NWS Technical Attachment SR 90-23 | Feels Like Temperature |
| Apparent temperature, for the band where neither NWS formula applies | Steadman, R.G. (1984); published operationally by the Australian Bureau of Meteorology as AT = Ta + 0.33e − 0.70ws − 4.00 | Feels Like Temperature |
| Vapour pressure from relative humidity | Australian Bureau of Meteorology, e = (RH/100) · 6.105 · exp(17.27Ta/(237.7+Ta)) | Feels Like Temperature |
| Solar term uses net radiation absorbed, not incident irradiance | Steadman radiation form, AT = Ta + 0.348e − 0.70ws + 0.70Q/(ws+10) − 4.25 | Feels Like Temperature |
| Frostbite time / heat-risk bands | NWS wind chill chart; NWS heat index guidance | Feels Like Temperature |
| The five constant-acceleration equations of motion (SUVAT) | Standard Newtonian kinematics — all five follow from a = dv/dt held constant, and appear in every introductory mechanics text with no single citable origin. Standard gravity is the CGPM defining value 9.80665 m/s². Solving for time uses a numerically stable quadratic; both roots are reported when both are non-negative, and both signs of a square root are reported when solving for a velocity. All ten three-known solve paths are cross-checked in suvat_model.py against 400 randomised motions each | SUVAT Calculator |
| Solar position, sunrise/sunset algorithm | Meeus, J., Astronomical Algorithms (2nd ed.) — standard reference for solar/lunar position calculations | Sun Phase |
| Twilight zenith angles: civil 96°, nautical 102°, astronomical 108° | Standard definitions — the sun 6°, 12° and 18° below the horizon respectively (US Naval Observatory / NOAA convention). Each is fed to the same sunrise routine with a different zenith, so all four boundaries come from one algorithm rather than four approximations. | Sun Phase |
| Golden hour taken as the sun between the horizon and 6° elevation | Widely used photographic convention. The page computes the real DURATION from the sun's angle rather than assuming an hour, because the name only holds in mid-latitudes: near the equator the sun crosses that band almost vertically (~28 min in Singapore) and at high latitude it crawls (~2h 34m at Longyearbyen in September). | Sun Phase |
| Peak solar altitude 90° − |latitude − declination|, and noon shadow = cot(altitude) | Exact spherical-geometry result for the sun's elevation at local solar noon; the shadow figure is its cotangent, i.e. metres of shadow per metre of height. Reported as a limit below ~3° elevation, where the cotangent runs away. | Sun Phase |
| Sunrise and sunset taken at 90.833° zenith: 34′ refraction plus 16′ solar semi-diameter | NOAA Solar Calculator; Meeus ch. 15, which gives the standard horizon altitude of −0°50′ | Sun Phase |
| Declination and equation of time from a Fourier series | Spencer, J.W. (1971), Fourier series representation of the position of the sun, as used by the NOAA solar calculator | Sun Phase |
| The true month between new moons runs from about 29.3 to 29.8 days around a mean of 29.530589 days; a new moon found from the mean can be out by up to ~14 h | IAU value for the mean; Meeus (1998), chapter 49. The mean is used here only to start the search for each phase: every moment is then found from the real positions of the Moon and the Sun. | Moon Phase |
| The phase of the Moon as the angle between the Moon and the Sun along the ecliptic (0° new, 90° first quarter, 180° full, 270° last quarter), and the lit fraction as (1 − cos e)/2 | Meeus (1998), chapters 48 and 49 (the illuminated fraction; the phases). The moments are found by Newton’s method on the angle, from the real positions of both bodies. | Moon Phase |
| All 248 new, quarter and full moons of 2018, 2021, 2023, 2026 and 2030 are found, the worst within 10 minutes; the average is 2.4 minutes late | US Naval Observatory, phases of the Moon, from its web service, compared in models/moon_model.py. The steady lag is the difference between the time scale of the series and universal time (about 69 seconds) and the series’ own error. | Moon Phase |
| The Moon’s distance runs from 356,400 to 406,700 km; its average closest approach is 363,300 km and its average farthest point 405,500 km; it is 33.5′ wide at its nearest and 29.4′ at its farthest | NASA Moon fact sheet. The widths are 2 arcsin(1,737.4 km / distance). A supermoon here is a full or new moon nearer than the average closest approach, a micromoon one farther than the average farthest point: a convention (the term is Richard Nolle’s, 1979), so lists differ. | Moon Phase |
| The 27 perigees and apogees of 2026: times good to about 6 hours and distances to about 500 km | Compared with a published table of the year, moonblink.info, in UT, centre to centre. The worst here is 5.6 hours and 513 km. | Moon Phase |
| Eclipses: 23 of the 24 solar and lunar eclipses of 2025 to 2029 found with the right type, the time within 5 minutes; the one missed is a penumbral eclipse of magnitude 0.001 | NASA, Five Millennium Catalog of Solar Eclipses and of Lunar Eclipses (Espenak and Meeus). Gamma is the distance of the shadow’s axis from the Earth’s centre in Earth radii, here the Moon’s distance times the sine of its smallest angle from the Sun. A central eclipse needs gamma under 0.9972 and a partial one under about 1.54 (Meeus, chapter 54). | Moon Phase |
| Magnitudes good to 0.003 for a total or annular eclipse and to about 0.06 for a deep partial one; the position is good to about 200 km on the ground, so a place cannot be placed inside the path of totality | The same comparison. A 2′ error in the Moon’s position is 2/60 ÷ 0.95 = 0.035 of an Earth radius, about 220 km at the Earth’s surface, and it moves the magnitude of a grazing partial eclipse by about 1.8 times that fraction. | Moon Phase |
| The Earth’s shadow at the Moon is taken as 2% wider than its geometric size, for the atmosphere | Danjon’s rule of 1/50, as used by Meeus (1998), chapter 54. Without it the umbral magnitudes are too small by about 0.03. | Moon Phase |
| The eclipse of 12 August 2026: London sees 91.4% of the Sun covered at 19:13 BST with the Sun 10.5° up; Reykjavik is in the path of totality at 17:48 GMT with the Sun about 25° up | Published local circumstances, Unistellar eclipse guide for London and Reykjavik. This tool gives 91.7% at 19:14 BST with the Sun 10.2° up, and a total eclipse in Reykjavik at 17:50 GMT. | Moon Phase |
| The equinoxes and solstices to within a quarter of an hour; the harvest moon is the full moon nearest the September equinox; a blue moon is the second full moon of a calendar month, a seasonal blue moon the third of four in a season, a black moon the second new moon of a month | Meeus (1998), chapter 27, and the US Naval Observatory’s 2026 times. The names (Wolf, Snow, Worm and so on) are the traditional American ones popularised by the Old Farmer’s Almanac; the date of a blue moon depends on the time zone, so it is taken in the place chosen. | Moon Phase |
| The Moon is moving away by about 3.8 cm a year | Lunar laser ranging to the Apollo and Soviet reflectors; Dickey et al. (1994), Science 265, 482. | Moon Phase |
| The Moon looks bigger near the horizon but is the same width high or low | The moon illusion: Ross and Plug, The Mystery of the Moon Illusion (Oxford University Press, 2002). The angular width changes by under 2% between horizon and zenith, because the observer is up to one Earth radius closer to it overhead. | Moon Phase |
| Arctic/Antarctic Circle latitude (66.56°) and the polar day/night terminator condition | Standard spherical astronomy — the terminator condition sin(elevation)=0 solved per latitude; the 66.56° threshold follows directly from Earth's 23.44° axial tilt (90° − tilt) | Sun Phase |
| The sun’s altitude and compass bearing from the hour angle, sin h = sin φ sin δ + cos φ cos δ cos H, and the bearing at sunrise cos A = (sin δ − sin φ sin h0)/(cos φ cos h0) | Meeus, J., Astronomical Algorithms, 2nd edition (Willmann-Bell, 1998), chapters 13 and 15 | Sun Phase |
| The equation of time, as the sun running up to 16 minutes ahead of or 14 minutes behind the clock, and solar noon on the clock as 12:00 + (zone offset − 4 × longitude) − equation of time + daylight saving | Meeus (1998), chapter 28; NOAA Global Monitoring Laboratory, Solar Calculation Details | Sun Phase |
| Time-zone offsets and daylight saving, read from the browser’s copy of the time-zone database rather than assumed | IANA Time Zone Database (tzdata); ECMAScript Internationalization API (ECMA-402) | Sun Phase |
| Day length at the equator on the equinox (12 h 07 min), London’s longest and shortest days (16 h 39 min and 7 h 50 min) and the earliest sunset in London (15:51, 13 December) | US Naval Observatory, Sun or Moon Rise/Set Table for One Year (aa.usno.navy.mil); timeanddate.com, Sun calculator for London 2026 | Sun Phase |
| IANA time zone database | IANA tz database, read live through the browser's Intl API — so zone rules and DST transitions are as current as the device. The world map's day/night terminator uses the NOAA solar-position approximation (Spencer's Fourier fit for declination and the equation of time), the same one behind Sun Phase, with the terminator taken as the locus where solar elevation is zero. Cross-checked in timezone_model.py against Python's own copy of the tz database. The clock-changes section hard-codes no country's rules: it samples each zone's offset daily through the year and pins every change to the minute, so it is exactly as current as the browser's data. Its two written notes are checked against tz data 2026c: Ireland moves on the UK's dates, and Morocco returned to UTC+0 for good on 20 September 2026 (Decree No. 2.26.530, Bulletin officiel, 29 June 2026), ending its Ramadan changes. A browser with older data still puts Morocco on UTC+1, and the page detects and says so. Exchange trading hours are each venue's published continuous session, excluding pre- and post-market; Tokyo, Shanghai and Hong Kong lunch breaks included. Session times change occasionally and holidays are not modelled — check the exchange for anything that depends on it | Time Zone Converter |
| Residential electricity rates by country | Ember / Our World in Data electricity price series — representative national averages, edit to match your own bill | Energy Cost Calculator |
| A UK electricity standing charge of roughly 60p/day, about £219 a year | Ofgem price cap standing-charge component; payable regardless of consumption, so it is 36% of a 1,500 kWh bill and 9% of a 9,000 kWh one | Energy Cost Calculator |
| Always-on appliances are entered as average power, not plate rating | Duty cycle folded into the wattage — a fridge at 55 W average against a ~150 W compressor plate. Verified in energy_model.py | Energy Cost Calculator |
| Grid carbon intensity by country | Ember Yearly Electricity Data | Energy Cost Calculator |
| Typical appliance wattages | U.S. DOE Energy Saver appliance energy guide — starting points, fully editable | Energy Cost Calculator |
| Solar panel output model, standard test conditions | IEC 61215 (1000 W/m², 25°C cell temperature) | Solar PV Estimator |
| Splitting monthly irradiation into beam and diffuse from the clearness index | Erbs, Klein & Duffie, Solar Energy — monthly-average correlation (the day-by-day form does not apply to a monthly mean) | Solar PV Estimator |
| In-plane irradiation on a tilted, oriented surface (isotropic sky) | Liu & Jordan; Duffie & Beckman, Solar Engineering of Thermal Processes, ch. 2 — POA = HbRb + Hd(1+cosβ)/2 + Hρ(1−cosβ)/2 | Solar PV Estimator |
| Ground reflectance taken as 0.20 | Duffie & Beckman, typical value for ordinary ground without snow | Solar PV Estimator |
| Everest / cruising altitude / Titanic / Challenger Deep reference points | National Geographic; NOAA Ocean Exploration (Challenger Deep depth, ~10,935 m). The atmosphere model is ISO 2533 / ICAO Standard Atmosphere in full — all seven layers to the 84,852 m ceiling the standard itself defines; the Kármán line at 100 km lies above it and is marked as such rather than clamped to fit | Altitude & Depth |
| A hot or cold day as ISA + ΔT, and density matching the standard again near 8–9 km | The International Standard Atmosphere (ICAO Doc 7488) with every layer shifted by the same ΔT and sea-level pressure held at 101,325 Pa, the convention aircraft performance data uses for non-standard days. Pressure is integrated hydrostatically with the shifted temperatures; the height where the density difference vanishes is the meteorologists’ isopycnic level, near 8 km (American Meteorological Society, Glossary of Meteorology). Checked by numerical integration in altitude_model.py: 8,720 m for ISA + 20 and about 8,140 m for ISA − 20 | Altitude & Depth |
| Claim | Source | Tool |
|---|---|---|
| Beam deflection formulas | Roark's Formulas for Stress and Strain; Euler-Bernoulli beam theory | Beam Deflection & Stress Calculator |
| Darcy friction factor (Colebrook / Swamee-Jain) | Swamee, P.K. & Jain, A.K. (1976), J. Hydraulics Division ASCE — explicit approximation of the Colebrook equation | Pipe Flow |
| Bolt torque-tension relationship | Shigley's Mechanical Engineering Design, ch. on threaded fasteners; K-factor method | Bolt Torque |
| Bolt property classes and proof stresses (metric) | ISO 898-1 — mechanical properties of carbon and alloy steel fasteners. Class 8.8 is diameter-dependent: proof 580 N/mm², yield 640, tensile 800 for d ≤ 16 mm, and 600/660/830 above. The tensile stress area As = π/4 (D − 0.9382P)² is from the same standard. | Bolt Torque |
| Bolt grades and proof loads (imperial) | SAE J429 — grade 2 is diameter-dependent (55,000 psi to ¾", 33,000 above); grades 5 and 8 hold across the sizes offered. | Bolt Torque |
| Nut factor K by finish and lubrication | Bickford, An Introduction to the Design and Behavior of Bolted Joints; standard published nut factors, with the ±25% scatter the page states | Bolt Torque |
| Otto/Diesel/Brayton/Carnot cycle efficiency | Cengel & Boles, Thermodynamics: An Engineering Approach — standard air-standard cycle analysis | PV Diagram |
| ISO 281 bearing life (L10) | ISO 281:2007 — rolling bearing dynamic load ratings and rating life | Bearing Calculator |
| O-ring groove dimensions | Parker O-Ring Handbook (ORD 5700); AS568 standard sizes | O-Ring Calculator |
| A fitted O-ring’s cord diameter is the free cord divided by √(1+stretch), so the squeeze a gland delivers is below the squeeze it was cut for | Follows from near-incompressibility: the cord’s volume is conserved, its length goes as the circumference, so cross-sectional area goes as 1/(1+s) and diameter as 1/√(1+s). This is a lower bound on the thinning — non-uniform stretch and a Poisson ratio just under 0.5 both remove more | O-Ring Calculator |
| Installed stretch is held to about 5%, and circumferential compression to about 3% | Customary O-ring gland design practice (AS568 / ISO 3601-2 usage, and manufacturer design guides): beyond ~5% stretch, stress relaxation and ozone attack shorten life sharply; beyond ~3% compression the ring buckles and twists in its groove rather than seating flat | O-Ring Calculator |
| Gland fill is checked at the service temperature using 2×10⁻⁴/K for the elastomer against 1.2×10⁻⁵/K for steel | Published linear expansion coefficients: filled elastomers fall in the 1.6–2.4×10⁻⁴/K band, carbon steel near 1.2×10⁻⁵/K — roughly a factor of fifteen, which is why the fill ceiling sits below 100%. Fluid swell is additional and is not modelled | O-Ring Calculator |
| Circlip / retaining ring dimensions | DIN 471 (external), DIN 472 (internal) standard series | Circlip Calculator |
| Retaining ring thrust load model | Groove capacity as bearing area × yield strength, the form published by Smalley and Rotor Clip. The 0.83 open-ring factor is derived from the published DIN load column, not quoted from a catalogue. Ring capacity is reproduced as published and is not computed. | Circlip Calculator |
| Structural steel yield strengths (S235/S275/S355) | EN 1993-1-1 §3.2.6 nominal values, t ≤ 40 mm | Circlip Calculator |
| ISO limits and fits tolerance grades | ISO 286-1/286-2 | Limits & Fits Calculator |
| Interference-fit contact pressure, press force, transmissible torque and shrink temperature | Lamé's thick-cylinder solution for a solid shaft in a hollow hub: p = δ / (d[(R+νo)/Eo + (1−νi)/Ei]) with R = (Do²+d²)/(Do²−d²); see Roark's Formulas for Stress and Strain, thick cylinders, and Shigley's Mechanical Engineering Design on interference fits. Both parts are compliant — treating the shaft as rigid, which is what copying the internal-pressure formula alone gives you, overstates an all-steel joint by 36%. Torque is the slip point at the stated friction with no safety factor, assuming clean dry surfaces and full contact; surface roughness flattened on assembly is not deducted, and on a small fit that is most of the interference. Verified in limitsfits_model.py. | Limits & Fits Calculator |
| Elastic moduli, Poisson's ratios and yield strengths by material | Representative values for one common grade of each material — ASM Engineered Materials Reference Book and standard engineering tables. Real stock varies with grade, temper and heat treatment, so these size a joint rather than certify one. Cast iron is listed by tensile strength rather than yield, and flagged brittle, because it cracks instead of taking a permanent set. | Limits & Fits Calculator |
| Thermal expansion coefficients by material | ASM Engineered Materials Reference Book | Limits & Fits Calculator |
| Unit conversion factors (imperial/metric) | NIST Handbook 44 Appendix C (customary–SI factors). SI base units: see the SI units guide row above (not Handbook 44). All 351 factors are generated by unit_model.py rather than typed: they are built from a handful of exact anchors (the inch is 0.0254 m, a US gallon is 231 cubic inches, a nautical mile is 1852 m, standard gravity is 9.80665 m/s²) and checked against definitional identities and against their own parent units, so area cannot drift from length squared. The historical temperature scales (Delisle, Newton, Réaumur, Rømer) are from their published fixed points; the unusual units behind the toggle are conventional values with no single standards body — the smoot and the beard-second are jokes with an agreed size, not measurements | Unit Converter |
| Fuel economy converts through kilometres per litre, and mpg ↔ L/100 km is a reciprocal: 1 mpg(US) × L/100km = 235.215, 1 mpg(UK) × L/100km = 282.481 | Derived from the definitional mile (1609.344 m), US gallon (3.785 411 784 L) and imperial gallon (4.546 09 L). Every fuel unit is round-tripped through km/L in models/unit_model.py, and both constants are computed there rather than typed | Unit Converter |
| The values marked on the relationship graph — water freezing and boiling, body temperature, −40, an inch, a mile, a pound, the 30 and 70 mph limits, one atmosphere | Each is checked in models/unit_model.py by converting it through the same table the converter uses, so a marked landmark cannot drift from the arithmetic beside it. −40 is the unique fixed point of the Celsius–Fahrenheit affine map | Unit Converter |
| Stopping distances, friction coefficients and reaction times | The official table is Highway Code Rule 126, reproduced exactly. Its assumptions are DERIVED from it rather than asserted: stopping_model.py reverse-engineers a reaction time of 0.671 s and a deceleration of 6.56 m/s² (0.669 g) from the published thinking and braking distances, then checks that feeding those back through v²/2a reproduces every published total within 1.3%. Braking distance is v²/(2g(μ·cosθ + sinθ)), so gradient enters as a component of gravity. Friction coefficients are mid-range published values for a car with ABS on a reasonable tyre — 0.9 dry, 0.6 wet, 0.25 packed snow, 0.12 ice — and vary widely in the literature with tyre, tread, temperature and road age; they should be read as comparisons, not promises. Vehicle factors and reaction times are representative figures, not measurements of any particular vehicle or driver. Not modelled: brake build-up, weight transfer, ABS cycling, fade, camber, tyre condition, or the long tail of the real reaction-time distribution. This is arithmetic about idealised friction and is not driving advice. | Stopping Distance Calculator |
| Nozzle flow: choking, shocks, separation and thrust | Quasi-one-dimensional isentropic flow of a calorically perfect gas with normal shocks. Every relation is verified in nozzle_model.py against the standard compressible-flow tables: p*/p₀ = 0.5283 and T*/T₀ = 0.8333 for air; A/A* = 1.6875 at Mach 2 and 4.2346 at Mach 3; normal-shock results at M₁ = 2 (M₂ = 0.57735, p₂/p₁ = 4.5, 28% of total pressure lost) and M₁ = 3. The three critical back-pressure ratios and the position of an internal shock are derived numerically rather than tabulated, and the area–Mach inversion is checked by round-trip. Choked mass flow reproduces the standard 0.0404·A*p₀/√T₀ coefficient for air. Separation uses the Summerfield criterion — a single pressure ratio near 0.4 standing in for a boundary-layer calculation, and quoted anywhere from 0.3 to 0.4. Gas properties are representative values; real rocket exhaust is not calorically perfect and its ratio of specific heats changes down the bell. Not modelled: two-dimensional flow, divergence loss, boundary layers, heat transfer, finite-rate chemistry. The plume is placed, not solved: see the next row. For understanding the behaviour, not for cutting metal | Nozzle Simulator |
| Engine figures: chamber pressure, expansion ratio, thrust and specific impulse for Merlin 1D and Merlin Vacuum, Raptor 2 and Raptor Vacuum, RS-25, F-1, RD-180, Vulcain 2 and RL10B-2 | Published figures as given on each engine's Wikipedia article (September 2026), which cite the manufacturers: SpaceX (Merlin 1D 9.7 MPa, 845 kN, 282 s; Merlin Vacuum 165:1, 981 kN, 348 s; Raptor 2 300 bar, 34.34:1, 2,256/2,530 kN, 347 s; Raptor Vacuum 80:1, about 380 s - SpaceX's own statements, not independent test data), Aerojet Rocketdyne (RS-25 20.64 MPa, about 69:1, 1,860/2,279 kN at 109%, 366/452.3 s, throat 0.262 m; RL10B-2 280:1, 110.1 kN, 465.5 s), Rocketdyne (F-1 70 bar, 16:1, 6,770/7,770 kN, 263/304 s), NPO Energomash (RD-180 26.7 MPa, 36.87:1, 3,830/4,150 kN, 311/338 s) and ArianeGroup (Vulcain 2 117.3 bar, 58.2:1, 1,359 kN, 429 s). Merlin 1D's 16:1 and the RL10B-2's chamber pressure of 644 psia (44.4 bar) are from secondary sources: the 16:1 is widely reported for the sea-level Merlin but is not in its Wikipedia infobox, and 644 psia is from the published test report on the RL10B-2 carbon-carbon nozzle extension. Derived, not published: throat and exit diameters, sized so the page's ideal thrust equals the published thrust; the RS-25's derived throat (0.27 m) is within 3% of its published 0.262 m. Chamber temperatures are representative values for each propellant, not engine figures | Nozzle Simulator |
| Why the 40% separation rule flags the RS-25 and Vulcain 2 at sea level, when both run full on the ground | R.A. O'Leary and J.E. Beck, Nozzle Design (Rocketdyne, spring 1992): the SSME's wall exit pressure was raised from 4.6 to 5.7 psia by shaping the end of the bell, at 0.1% efficiency, while the core flow is near 2 psia. R. Stark, Flow Separation in Rocket Nozzles - an Overview (DLR): the Vulcain 1 nozzle was designed for a wall exit pressure of about 0.4 bar, per Summerfield. The page's one-dimensional exit pressure is a cross-section average and cannot see the wall-to-core difference | Nozzle Simulator |
| Methalox exhaust: γ = 1.20, R = 400 J/(kg·K) | A representative frozen value for methane and oxygen near the mixture ratios of current engines (molar mass about 21 g/mol). With one fixed γ, ideal theory comes out below Raptor's figures, as it does for the kerosene-burning RD-180: at 267-300 bar the exhaust recombines as it expands and returns heat to the flow, which a calorically perfect gas cannot represent. The page says so on those engines' cards rather than tuning the gas to match | Nozzle Simulator |
| Bell contour, shock-cell spacing and Mach-disc position | The bell is a Rao thrust-optimised parabola: a 1.5 Rt entrant arc, a 0.382 Rt throat arc, then a quadratic curve between Rao's throat and exit wall angles, interpolated from his published charts for 60, 80 and 90% bells (G.V.R. Rao, Jet Propulsion, 1958). Shock-cell length is Pack's 1.306 Dj√(Mj²−1), with Dj and Mj the fully-expanded jet diameter and Mach number (D.C. Pack, 1950). An under-expanded jet's Mach disc sits at x/De = 0.69 Me√(γpe/pa) (Lewis & Carlson, AIAA Journal, 1964), once pe/pa passes about 2; an over-expanded jet gets one when its lip shock cannot reflect regularly off the axis. The Prandtl–Meyer and oblique-shock limits are checked in nozzle_model.py against NACA Report 1135. The plume's brightness, colour and turbulence are drawn, not computed | Nozzle Simulator |
| Spring rate & Wahl stress-correction formula | Shigley's Mechanical Engineering Design, ch. on mechanical springs; Wahl, A.M. (1944), Mechanical Springs | Spring Rate Calculator |
| Common spring wire material properties | SAE J1123/J1124; Associated Spring/Barnes Group design handbook | Spring Rate Calculator |
| Spring surge (natural) frequency, both ends seated | A compression spring between two seats is a fixed-fixed elastic continuum, so its fundamental is the bar result f₁ = c/2L, reducing to ½·√(k/m) with the full active-coil mass — as given for springs between parallel plates in Shigley’s Mechanical Engineering Design, ch. on mechanical springs. spring_model.py derives it twice (from k and m, and in closed form) and requires the two to agree | Spring Rate Calculator |
| The 13× surge margin guidance | Standard spring-design guidance that the surge frequency should be at least about thirteen times the forcing frequency; quoted in Shigley’s and in SMI (Spring Manufacturers Institute) design practice. It is a chosen threshold, not a derived limit — dual springs and progressive pitch deliberately break the resonance up, so a single-frequency check understates a well-designed valve train | Spring Rate Calculator |
| Solid height and buckling slenderness thresholds | Solid height = wire diameter × total coils for squared-and-ground ends. Buckling slenderness L₀/D guidance of about 5.2 between parallel plates and about 2.6 with an end free to pivot, per Shigley’s treatment of spring stability. Both are geometry limits reached before any stress limit | Spring Rate Calculator |
| Euler buckling & Johnson's parabolic formula | Timoshenko, S.P., Theory of Elastic Stability; Shigley's Mechanical Engineering Design, ch. on columns | Column Buckling Calculator |
| Imperfection sensitivity: δ = e0/(1 − P/Pe) and the Perry–Robertson first-yield load | Robertson, A., "The strength of struts", ICE Selected Engineering Papers No. 28 (1925); Timoshenko, Theory of Elastic Stability, on the beam-column with initial curvature. The same formulation underlies the buckling curves of BS 5950-1 and Eurocode 3, where the imperfection factor is calibrated to test data rather than to a single assumed bow. Initial bow is taken as a fraction of length; L/1000 is representative of rolling tolerance for hot-rolled sections. | Column Buckling Calculator |
| Structural material yield strengths | ASTM A36 (steel), ASTM B209 (aluminium) material standards | Column Buckling Calculator |
| Gear-mesh kinematics (ratio, speed, torque, mesh efficiency) | Shigley's Mechanical Engineering Design, ch. on spur/helical gears — standard gear-train kinematics | Gear Train Calculator |
| Planetary (epicyclic) ratios from the Willis equation; planet tooth count and equal-spacing assembly constraint | Willis relation (ωs−ωc)/(ωr−ωc) = −Nr/Ns as given in Shigley's Mechanical Engineering Design and Norton, R.L., Design of Machinery, ch. on epicyclic trains. The three closed-form arrangements, the two-degrees-of-freedom residual and the 24/72 textbook case are each re-derived and checked in gear_model.py before reaching this page | Gear Train Calculator |
| Pitch diameter d = m·z, centre distance m(z₁+z₂)/2, and the 2/sin²α undercut limit | ISO 53 / ISO 54 standard basic rack and module series; undercut limit is the standard full-depth result quoted in Shigley's and Norton — 17.1 teeth at 20°, 31.9 at 14.5°, 11.2 at 25°, each verified in gear_model.py. It is a geometry limit on the generating cutter, not a strength criterion, and profile shift defeats it | Gear Train Calculator |
| Slider-crank piston kinematics; 4-stroke cam/valve timing; 2-stroke piston-port scavenging sequence | Slider-crank kinematics x(θ)=r·cosθ+√(L²−r²sin²θ) — Norton, R.L., Design of Machinery; engine cycle and valve-event treatment follows Heywood, J.B., Internal Combustion Engine Fundamentals. Port timing represents a typical piston-ported, crankcase-scavenged 2-stroke layout, not any single manufacturer's spec | Engine Cycle Simulator |
| Ideal air-standard efficiencies: Otto 1 − r1−γ; Diesel the same with a cutoff bracket that is always > 1; Atkinson 1 − γ(T₄−T₁)/(T₃−T₂) with complete expansion | Standard cold-air-standard cycle analysis with γ = 1.4. The page builds the corner states and takes the efficiency from the AREA of the loop it draws; models/engine_model.py reproduces that integration and checks it against these closed forms for all three cycles, so the diagram and the number cannot disagree | Engine Cycle Simulator |
| A Wankel rotor turns at one third of eccentric-shaft speed, so one chamber’s cycle spans 1080° of shaft — yet each rotor fires once per shaft revolution | Fixed by the geometry of the design: a three-lobed rotor on a 3:1 eccentric, whose apexes generate the epitrochoid housing. Three faces against a 3:1 reduction cancel exactly | Engine Cycle Simulator |
| Mean piston speed limits (20 m/s production, 20–25 competition, ~25 for cast pistons) | Heywood, J.B., Internal Combustion Engine Fundamentals — mean piston speed as the governing design limit. Band figures cross-checked against published engine-design references; the Formula 1 anchor is the FIA regulation 80 mm bore × 53 mm stroke, which gives 26.5 m/s at the 15,000 rpm limit by 2 × stroke × rpm. | Engine Cycle Simulator |
| BMEP ranges by engine type (8–12 bar NA petrol, up to 25 turbo diesel) | Published brake-mean-effective-pressure ranges for naturally aspirated and turbocharged petrol and diesel engines. BMEP is an input on this page rather than a derived figure — it stands in for the breathing and combustion the model does not simulate. | Engine Cycle Simulator |
| Ideal air-standard Otto efficiency 1 − r1−γ | Standard cold-air-standard cycle analysis at γ = 1.4 (Cengel & Boles, Thermodynamics). Shown as a ceiling no real engine reaches, not a prediction — the same treatment the PV Diagram tool uses. | Engine Cycle Simulator |
| Hydraulic force (Pascal's law, F = P·A) | Standard fluid statics — Pascal's principle, cross-referenced across multiple fluid mechanics references | Hydraulic Force Calculator |
| Claim | Source | Tool |
|---|---|---|
| Fluid density and viscosity reference values | NIST Chemistry WebBook (water); CRC Handbook of Chemistry and Physics; ISO 3448 viscosity grades; EN 590 (diesel) | Fluid Properties |
| Viscosity vs temperature for oils: log₁₀log₁₀(ν+0.7) = A − B·log₁₀T | ASTM D341, Standard Practice for Viscosity-Temperature Equations and Charts for Liquid Petroleum Products | Fluid Properties |
| Water viscosity to within 2% over 0–100 °C | Vogel–Fulcher–Tammann form as tabulated in NIST/IAPWS reference data for liquid water | Fluid Properties |
| Water density, and its maximum at 3.98 °C | Kell, G. S., Journal of Chemical & Engineering Data (density of pure water 0–150 °C) | Fluid Properties |
| Viscosity Index is not computed because D2270 needs tabulated L and H | ASTM D2270, Standard Practice for Calculating Viscosity Index (polynomial form valid only above ν₁₀₀ = 70 cSt) | Fluid Properties |
| Gas density from the ideal gas law with a second-virial correction, ρ = pM/(ZRT), Z = 1 + Bp/(RT) | B from critical temperature, critical pressure and acentric factor by the Pitzer–Abbott correlation (Smith, Van Ness & Abbott, Introduction to Chemical Engineering Thermodynamics). Critical constants from the NIST Chemistry WebBook and Poling, Prausnitz & O’Connell; air as a pseudo-pure fluid (Lemmon et al., 2000). Checked against the CRC Handbook densities of eight gases at 0 °C and 101.325 kPa, all within 0.05% | Air & Gas Density |
| Where a gas would condense, and humid air | Lee–Kesler vapour-pressure correlation on the same critical constants; for water, Buck’s equation in its 1996 revision of Buck (1981), Journal of Applied Meteorology 20, 1527. Humid air as dry air and water vapour, each an ideal gas at its own partial pressure, which reproduces the published 1.194 kg/m³ for saturated air at 20 °C | Air & Gas Density |
| Speed of sound √(γRT/M) and gas viscosity by Sutherland’s law | Sutherland constants from White, Viscous Fluid Flow, Table 1-2, used only for the gases it covers (air, N₂, O₂, CO₂, Ar, H₂, steam); no viscosity is shown for the others. Ratios of specific heats are held constant at their room-temperature values | Air & Gas Density |
| The ideal gas constant R = NAkB = 8.314462618 J/(mol K), with NA = 6.02214076×1023 and kB = 1.380649×10−23 J/K, exact since 2019; and the molar volume of an ideal gas, 22.41397 L at 273.15 K and 101.325 kPa and 22.71095 L at 273.15 K and 100 kPa | NIST CODATA fundamental physical constants. The model checks the page’s constants and molar volumes against scipy.constants (CODATA) and every unit against its definition (the pound, the foot, the standard atmosphere). | Ideal Gas |
| The molar masses of helium, neon, argon, hydrogen, nitrogen, oxygen, carbon dioxide, methane, ammonia, chlorine, propane and sulfur hexafluoride | The standard atomic weights of the IUPAC Commission on Isotopic Abundances and Atomic Weights, added up; the model compares all twelve with the atomic weights to 0.025 percent (the weights of hydrogen, carbon, chlorine and sulfur are intervals). | Ideal Gas |
| The van der Waals constants a and b of the twelve gases (for example nitrogen 1.370 L² bar/mol² and 0.0387 L/mol), from which the critical point Tc = 8a/27Rb, Pc = a/27b2 and the Boyle temperature a/Rb follow | Weast, CRC Handbook of Chemistry and Physics (53rd edition), as tabulated in the list of van der Waals constants. The model checks each pair against the measured critical constants of the gas (CRC Handbook) through a = 27R2Tc2/64Pc and b = RTc/8Pc, and the equation against its definition, the roots of its cubic, Zc = 3/8 and the second virial coefficient b − a/RT. | Ideal Gas |
| The Maxwell–Boltzmann distribution of speeds of the molecules of a gas, the most probable, mean and root-mean-square speeds √(2kT/m), √(8kT/πm) and √(3kT/m), the share faster than a speed, the pressure (1/3)(N/V)m⟨v2⟩ and the rate of effusion P/√(2πmkT) | The kinetic theory of gases (Atkins, Physical Chemistry, chapter on the kinetic model of gases). The model compares the density, the cumulative share, the survival function and the quantile of the speeds for the twelve gases at five temperatures with scipy.stats.maxwell (agreement to 10−9), integrates the density numerically for the means and the share above a speed, and sums the pressure over the speeds drawn. | Ideal Gas |
| The ideal gas law and its four special cases, Boyle’s law (PV constant), Charles’s law (V/T), Gay-Lussac’s law (P/T) and Avogadro’s law (V/n) | Any text of physical chemistry; the model puts each change of state back into PV = nRT and checks the combined law, the extrapolation of Charles’s law to zero volume at 0 K, and the four unknowns of the law found from the other three at 200 random states (agreement to 10−12). | Ideal Gas |
| Standard concrete mix ratios by grade (C20 1:2:4, C25 1:2:3, C30 1:1.5:3), treated as ratios by volume, and the 1.54× dry-volume correction | Cross-referenced across multiple published concrete-calculator and construction references. Splitting the dry volume by the ratio and converting each share at its own bulk density (cement 1440, sand 1600, aggregate 1500 kg/m³) gives 317 kg of cement per m³ for C20, inside the 300–330 kg/m³ band published for nominal 1:2:4, and a wet density of 2,499 kg/m³. The three materials sum to the quoted dry volume exactly | Concrete Mix Calculator |
| Strength falls roughly exponentially with the water-cement ratio — 0.60 keeps about 74% of a 0.50 mix and 0.70 about 55% | Abrams' law, S = A/B^(w/c), fitted to a published 28-day cube-strength table (0.40→50, 0.50→40, 0.60→30, 0.70→22 MPa); the fit tracks that table within 8% across the range. Shown only as a ratio against 0.50, because the absolute constants move widely with cement type, aggregate and compaction | Concrete Mix Calculator |
| Strength gain over time, and the effect of temperature on it — 70% takes about 4.8 days at 20°C but 9.5 days at 5°C | Eurocode 2 strength-development expression βcc(t) = exp{s[1 − √(28/t)]} with s = 0.25 for class N cement, combined with Nurse–Saul equivalent age on a −10°C datum. Reproduces the familiar textbook figures: 50% at about two days at 20°C, and 70% at seven days at 10°C | Concrete Mix Calculator |
| Atomic number, symbol, standard atomic weight, category, period, group, ground-state electron shell occupancies, density at STP, melting and boiling points, Pauling electronegativity and empirical atomic radius for all 118 confirmed elements | Values follow IUPAC standard atomic weights where an element has one, and the mass number of the longest-lived isotope where it does not — the page marks which. The dataset is held in periodic_model.py and validated against the periodic law before being emitted for the page: shell occupancies must sum to the atomic number, no shell may exceed its 2n² capacity, the period must equal the occupied-shell count (palladium alone excepted, being [Kr]4d10), group membership must match category, and the electronegativity, atomic-radius, density and melting-point trends must hold in the numbers — 96 checks. That validation established four genuine standard-atomic-weight inversions (Ar/K, Co/Ni, Te/I and Th/Pa) rather than the three usually quoted. The electron-shell diagram is a Bohr-style schematic of shell occupancy, not a depiction of orbitals. Properties above atomic number 100 are largely predicted rather than measured; isotopes, ions and allotropes are not modelled | Periodic Table |
| Molecular geometry, bond angles, molar masses and bond polarity | Geometry is not stored. Each molecule is held as a connectivity list and the three-dimensional shape is relaxed by mutual repulsion of bonding and non-bonding electron domains on a unit sphere — VSEPR (Gillespie and Nyholm) expressed as a minimisation rather than a lookup. It reproduces 109.47° tetrahedral, 120° trigonal planar, 90° octahedral and the 90/120° split of a trigonal bipyramid without any of those figures appearing in the source. Lone pairs are weighted 1.20× a bonding pair; that single figure is fitted across ammonia and water and leaves each about one degree from its measured angle (107.8° and 104.5°), which the page states per molecule rather than hiding. Molar masses and electronegativity differences are computed from the same generated dataset as the Periodic Table, so glucose returns 180.16 g/mol from its own atom count; hydrogens are placed by filling unsatisfied valences. C₆₀ is generated as a truncated icosahedron from the golden ratio and checked for 60 vertices, 90 edges and uniform degree 3. Bond lengths are sums of covalent radii — a few per cent, not spectroscopy. Bond-type cutoffs (0.4 and 1.7 Δ-electronegativity) are conventional and the page notes that they misfile hydrogen fluoride as ionic. Not modelled: conformers, formal charge, resonance beyond alternating bonds, or crystal packing past a small fragment | Molecules |
| The acid dissociation constants (pKa at 25 °C, in dilute solution) of the 27 acids and bases: strong acids and bases are taken as completely ionised, and weak bases are given as the pKa of their conjugate acid (pKa + pKb = 14) | The CRC Handbook of Chemistry and Physics (84th edition), as printed in the appendix of selected acid dissociation constants, and Perrin (Ionisation Constants of Inorganic Acids and Bases in Aqueous Solution, IUPAC), Haynes (CRC) and Harris as quoted on the Wikipedia pages of the substances, and the Engineering ToolBox. The model compares 47 of them with a published value in each of two tables, and none is more than 0.1 away. | Acids & Bases |
| The pH is the solution of the charge balance, [H+] + (strong cations − strong anions) = [OH−] + the charge of each weak acid or base in the form it is in at that pH, with no assumption that the ionisation is small and with the water’s own ions included | The systematic treatment of equilibrium (Harris, Quantitative Chemical Analysis, chapter 6; Butler, Ionic Equilibrium). The model solves the same equations in 60-digit decimal arithmetic with a separate root finder for 216 substance and concentration pairs and 400 random titration points (agreement to 10−14 in pH), and against the closed forms: the cubic for a weak acid, the quadratic, and the strong acid with water. | Acids & Bases |
| The ionisation of water: pKw falls from 14.95 at 0 °C to 14.00 at 25, 13.28 at 50 and 12.25 at 100, so pure water is neutral at pH 7.47, 7.00 and 6.12 | Bandura and Lvov (2006), The ionization constant of water over wide ranges of temperature and density, J. Phys. Chem. Ref. Data 35, 15, as tabulated in the article on the self-ionization of water, and the usual table of pKw against temperature; the model compares the values at 0, 25, 50, 75 and 100 °C with the first to 0.025. | Acids & Bases |
| The transition ranges of the seven indicators (methyl orange 3.1–4.4, bromocresol green 3.8–5.4, methyl red 4.4–6.2, bromothymol blue 6.0–7.6, phenol red 6.4–8.0, phenolphthalein 8.3–10.0 and thymolphthalein 9.3–10.5) and their colours | The usual laboratory table, as listed in the article on pH indicators. The model checks the ranges, and which indicator suits a strong acid with a strong base, a weak acid with a strong base, a strong acid with a weak base and a weak acid with a weak base, against the textbook answers. | Acids & Bases |
| The Henderson–Hasselbalch equation, pH = pKa + log([A−]/[HA]), and the buffer capacity β = 2.303 (h + oh + C α(1 − α)), greatest at pH = pKa where it is 2.303 C/4 | Van Slyke (1922), J. Biol. Chem. 52, 525, on the buffer value; the model compares the page with the exact capacity at fifteen points and with the equation wherever the smaller form is over a hundred times the hydrogen and hydroxide ions (agreement to 0.01 in pH). | Acids & Bases |
| Rain is about pH 5.6: carbon dioxide at 420 ppm in water, with a Henry’s law constant of 3.3×10−4 mol/(m3 Pa) and pKa1 = 6.35 for carbonic acid together with dissolved carbon dioxide | Sander (2015), Compilation of Henry’s law constants (version 4.0) for water as solvent, Atmos. Chem. Phys. 15, 4399; the model gives pH 5.64. The other everyday values on the pH scale (black coffee about 5, milk 6.7, blood 7.4, sea water 8.1, milk of magnesia 10.5, bleach 12.5) are typical figures from the USGS Water Science School and standard texts and are not computed here; vinegar, stomach acid, baking soda solution, household ammonia and drain cleaner are. | Acids & Bases |
| Electron configuration, state at 20 °C, discovery year and discoverer, what each name means, and first ionisation energy | Configuration and state are derived, not transcribed. The configuration is filled in Madelung order with the twenty measured exceptions listed explicitly (chromium, copper, palladium, the awkward corners of the d- and f-blocks), then written by increasing n and l as convention requires — filling order and writing order differ, which is why iron is [Ar] 3d6 4s2. It is checked by collapsing every subshell back onto the shell occupancies already in the dataset, so two independently sourced descriptions of each atom must agree for all 118. State comes from the melting and boiling points against 293.15 K, which reproduces the eleven gases and two liquids without asserting them; it is not applied above Z = 100, where the points are predicted and flerovium’s would otherwise put it beside mercury. Discovery years, discoverers and name etymologies follow IUPAC and standard histories of chemistry; first ionisation energies are NIST values, calculated rather than measured above fermium. Naming is often disputed — the year given is the commonly credited isolation or identification, not a claim about priority | Periodic Table |
| Five representative uses and five common compounds for each element | Selected as things a reader can recognise rather than an exhaustive list — an element with hundreds of industrial applications is shown the five that best explain why it is mined or made. Compounds are the commonest ones an element appears in, given as formula and name; a handful are named materials rather than formulas (PTFE, PVC, cisplatin, ATP). Held in periodic_model.py, where a check confirms every listed compound actually contains the element it is filed under, and that the 26 elements shown none are exactly the light noble gases and the elements too short-lived for chemistry. Not a claim about abundance or commercial volume | Periodic Table |
| The standard atomic weights of the 52 elements the tool knows, in the conventional values the IUPAC publishes for everyday use (for example hydrogen 1.008, carbon 12.011, oxygen 15.999, chlorine 35.45) | IUPAC Commission on Isotopic Abundances and Atomic Weights, standard atomic weights (2024); a copy of the table is kept in models/, and the model checks every weight lies within the CIAAW interval or uncertainty, to the rounding of the figures shown. | Moles & Solutions |
| The Avogadro constant 6.02214076×1023 mol−1, exact, and the molar volume of an ideal gas RT/P: 22.41397 L at 273.15 K and 101.325 kPa, 22.71095 L at 273.15 K and 100 kPa | NIST CODATA fundamental physical constants; the model compares the page with scipy.constants. | Moles & Solutions |
| The strengths and densities of the laboratory stock reagents: hydrochloric acid 37 percent, 1.18 g/mL; sulfuric acid 98 percent, 1.84; nitric acid 70 percent, 1.41; ammonia solution 28 percent, 0.90; glacial ethanoic acid 99.8 percent, 1.05; sodium hydroxide 50 percent, 1.52; phosphoric acid 85 percent, 1.69 | The usual strengths of the bottles from the major suppliers and the concentrated-solution tables of the CRC Handbook. They are typical: a bottle states a range, and the model checks only that the molarity from the formula 10 × percent × density / molar mass agrees with the commonly quoted molarities (12, 18, 16, 15, 17, 19, 15 mol/L) to 3 percent. | Moles & Solutions |
| The balancing of an equation as the whole-number null vector of the element-by-species matrix, and the standard atomic weights from which the molar masses are made | IUPAC Commission on Isotopic Abundances and Atomic Weights for the atomic weights; the balancing is linear algebra (the conservation of each element). The model compares the balancing with scipy’s null space and checks atom conservation on every example and 300 random reactions built from known coefficients. | Reacting Masses & Yield |
| The limiting reagent, the theoretical and percentage yield and the atom economy, from the balanced equation and the molar masses (for example 28 g of nitrogen and 8 g of hydrogen make at most 34.04 g of ammonia; the atom economy of iron from Fe2O3 + 3 CO is 45.8 percent) | Standard stoichiometry and the definition of atom economy (Trost, 1991, Science 254, 1471; the percentage by mass form used in school and industry). The model checks the amounts against direct arithmetic for 300 random reactions and the textbook examples. | Reacting Masses & Yield |
| Claim | Source | Tool |
|---|---|---|
| Resistor 4-band colour code | IEC 60062 — colour coding of resistance and capacitance values | Resistor Colour Code Calculator · also Electrical Circuit |
| E-series preferred values (E6, E12, E24, E96) are geometric progressions stepping by 10^(1/n) per decade, chosen so consecutive tolerance bands overlap — which is why 4.7k exists and 5k does not | IEC 60063 preferred number series. Verified: each published series tracks 10^(1/n) to within 5%, and the ideal step is smaller than the ±tolerance span ratio (1+t)/(1−t) for E6 at 20%, E12 at 10% and E24 at 5%, so the bands overlap by design | Resistor Colour Code |
| The published values are rounded, and rounding opens small real gaps in every series — E12’s worst step is 1.25 (12 to 15) against a ±10% span of 1.222 | Measured directly from the published value tables rather than from the idealised step: the worst consecutive ratio exceeds the tolerance span in E6, E12, E24, E48 and E96 alike. A 1.34k target sits more than 10% from both 1.2k and 1.5k, so no single E12 part reaches it | Resistor Colour Code |
| SMD marking: 3-digit (473 = 47k), 4-digit (4702 = 47k), R for the decimal point (4R7 = 4.7Ω), and EIA-96 (01C = 10k, 68X = 49.9Ω) | Standard surface-mount resistor marking conventions. EIA-96 uses a two-digit index into the E96 table followed by a letter multiplier (Z 0.001 through F 100000), so code 68 is E96 value 499 and X multiplies by 0.1 | Resistor Colour Code |
| Standard power ratings and 2× derating — 5 V across 220Ω dissipates 0.114 W and wants a 1/4 W part, not the 1/8 W the arithmetic alone allows | P = V²/R, against the standard rating ladder (1/16, 1/10, 1/8, 1/4, 1/2, 1, 2, 3, 5 W). The 2× factor is a working allowance rather than a standard: manufacturer ratings assume free air at 25°C, which an enclosure does not provide | Resistor Colour Code |
| Wire gauge current-carrying capacity (advisory only) | Rough illustrative bands for the circuit demo only — not a full NEC ampacity lookup. For design values use the dedicated Wire Gauge Calculator, which implements NEC Table 310.16 / Ch.9 Table 8 / 240.4(D) as cited there | Electrical Circuit |
| UK DNO reactive-power charging threshold (0.95 PF) | UK Power Networks connection terms; DNO reactive-power methodology statements | Power Factor Calculator |
| Capacitor sizing formula, kVAR = kW·(tanφ₁−tanφ₂) | Power-triangle relationship kVAR = kW·(tanφ₁−tanφ₂) — IEEE Std 141 (Red Book), Recommended Practice for Electric Power Distribution for Industrial Plants, power-factor correction chapter | Power Factor Calculator |
| Typical uncorrected equipment power factors (motors, lighting, welding) | Illustrative reference ranges compiled from multiple electrical-engineering references — genuinely equipment- and load-dependent, not fixed constants | Power Factor Calculator |
| Ideal transformer turns-ratio and ampere-turns relationships | Ideal-transformer relationships V₁/V₂ = N₁/N₂ and N₁I₁ = N₂I₂ — Fitzgerald, Kingsley & Umans, Electric Machinery, transformer chapter | Transformer Calculator |
| Primary current is output power divided by efficiency and primary voltage | Conservation of energy: P₋ₙ = Pₒₖₜ/η. The ideal form load/V₁ understates it by up to 12% on a small transformer at light load | Transformer Calculator |
| Efficiency peaks where copper loss equals fixed core loss | Standard result for a two-component loss model; the peak therefore sits below full load | Transformer Calculator |
| Transformer full-load efficiency by size class (94-99.75%) | Compiled from multiple transformer manufacturer and industry engineering references; distribution transformers 97-99%, large power transformers up to 99.75% | Transformer Calculator |
| Maximum transformer efficiency occurs where copper loss equals core loss | Standard transformer loss theory, confirmed across multiple power-systems engineering references — typically 40-70% of rated load for distribution transformers | Transformer Calculator |
| NEC Table 310.16 copper ampacity (60/75/90°C) and Chapter 9 Table 8 conductor resistance | National Electrical Code (NFPA 70), cross-referenced across multiple published NEC ampacity/resistance charts | Wire Gauge Calculator |
| NEC 240.4(D) small-conductor overcurrent protection limits (15A/20A/30A for 14/12/10 AWG) | National Electrical Code (NFPA 70) — overrides raw ampacity-table values for these three gauges | Wire Gauge Calculator |
| Ambient-temperature correction factors for conductor ampacity | NEC 310.15(B)(1). The closed form the published table is built from — correction = √((Tc − Ta) / (Tc − 30)) — is used here rather than a lookup, so there is no interpolation argument at a range boundary. wire_model.py asserts it reproduces fourteen published rows across the 60, 75 and 90 °C columns exactly | Wire Gauge & Voltage Drop Calculator |
| Adjustment factors for more than three current-carrying conductors | NEC 310.15(C)(1) — 80% for 4–6 conductors, 70% for 7–9, 50% for 10–20, 45% for 21–30, 40% for 31–40, 35% above. Applied multiplicatively with the ambient correction, both verified in wire_model.py | Wire Gauge & Voltage Drop Calculator |
| Terminal temperature limitation (why 90 °C cable is capped at the lug rating) | NEC 110.14(C). Derating is applied to the conductor’s own insulation column and the result is then capped at the terminal column — doing it in the other order derates the cable twice and makes 90 °C insulation look pointless when it is exactly what rescues a hot or crowded run | Wire Gauge & Voltage Drop Calculator |
| Claim | Source | Tool |
|---|---|---|
| Regional mortgage model differences (compounding, rate reversion, mortgage insurance, escrow) | Market conventions rather than a single citation: Canadian semi-annual compounding is required by the Interest Act; UK and Canadian lending fixes for 2–5 years against a longer amortisation; US PMI cancels at 80% LTV under the Homeowners Protection Act while Canadian CMHC and Australian LMI premiums are capitalised into the loan. Rates shown per market are indicative typical new lending, not quotes | Mortgage Calculator |
| UK stamp duty (SDLT) bands and first-time buyer relief | England & Northern Ireland residential rates as at April 2025 — 0% to £125k, 2% to £250k, 5% to £925k, 10% to £1.5m, 12% above; first-time buyer relief 0% to £300k then 5% to £500k, withdrawn entirely above £500k. Thresholds change with budgets and differ in Scotland (LBTT) and Wales (LTT); the tool says so | Mortgage Calculator |
| Affordability tests by market (LTI, DTI, GDS, debt-service ratio) | Typical lender or regulatory limits: UK loan-to-income around 4.5×, US debt-to-income around 36%, Canadian gross debt service around 39%, and the French HCSF 35% debt-service cap. Individual lenders apply their own stress tests, so the tool presents these as a rule of thumb rather than a decision | Mortgage Calculator |
| Fee drag modelled period-by-period on the balance | Method rather than a citation: charges are levied pro rata each compounding period on the opening balance, which is how platform and fund fees accrue. There is no closed form once fees and contributions interact, so the tool runs the schedule. The reported cost is the difference against the same plan with no charge, which is larger than the fees paid because the money taken would also have compounded | Compound Interest Calculator |
| Exact doubling time is ln 2 / ln(1+r); the Rule of 72 is a fitted approximation accurate near 8% | Standard result. Error changes sign at about 8%: +2.8% at 2%, −5.3% at 20% | Compound Interest Calculator |
| Future value of an annuity, used to verify the period loop | FV = P(1+i)ⁿ + PMT·[((1+i)ⁿ−1)/i], ×(1+i) for payments in advance — standard actuarial form | Compound Interest Calculator |
| Long-run reference returns marked on the rate slider | Rounded long-run nominal annual averages for orientation, not forecasts: cash ~2%, inflation ~3%, bonds ~4.5%, global equities ~7%, US equities ~10%. Realised returns over any particular period differ substantially from long-run averages | Compound Interest Calculator |
| Compound interest formula | Time-value-of-money compounding A = P(1 + r/n)^(nt) — Brealey, Myers & Allen, Principles of Corporate Finance, time-value chapter | Compound Interest Calculator |
| Mortgage amortisation formula | Standard fixed-rate amortisation mathematics; matches the method used by CFPB-regulated lenders | Mortgage |
| Car loan / hire-purchase amortisation; PCP balloon (optional final payment) | Same fixed-rate amortisation as mortgages; UK product shapes follow common FCA-regulated HP/PCP structures (deposit + amortising credit ± guaranteed future value). Illustrative only — not a regulated quote. | Car Finance |
| Currency conversion from ECB reference rates | Frankfurter open API publishing European Central Bank euro foreign exchange reference rates — mid-market, not a retail quote | Currency Converter · also Unit Converter Currency category |
| The UAE dirham and the Saudi riyal are fixed to the US dollar, at 3.6725 and 3.75 per dollar | The dirham has been pegged by the Central Bank of the UAE at 3.6725 per dollar since 1997, and the riyal by the Saudi Central Bank at 3.75 since 1986. The ECB does not publish either, so the page derives them from the dollar rate and labels them fixed | Currency Converter |
| "4% rule" safe withdrawal rate | Bengen, W.P. (1994), Journal of Financial Planning — the original Trinity-study-adjacent safe withdrawal rate research | FIRE Calculator |
| UK National Living Wage / National Minimum Wage rates | GOV.UK, Low Pay Commission — current as of April 2026 | Salary Converter |
| US federal minimum wage | U.S. Department of Labor — $7.25/hr, unchanged since 2009 | Salary Converter |
| Regional tipping norms (US/Canada 18–25%, UK service charge often included, Western Europe optional 5–10%, Australia/NZ ~10%, Japan/South Korea not customary) | Cross-referenced across multiple 2026 sources: Autopilot Travel's tipping guide, Matador Network, Capital One, Global Rescue, and World Population Review's country tipping rankings — individual figures vary a few points by source, the broad regional pattern does not | Tip Calculator |
| US federal/FICA and UK income tax/NI bands | IRS; GOV.UK income tax rates — simplified, single-filer approximation, not personalised tax advice | Salary Converter |
| Scottish income tax bands for 2026/27, and the 2026 Social Security wage base | Scottish Government, rates and bands 2026 to 2027 (starter to £16,537, basic to £29,526, the rest unchanged); SSA announcement, October 2025 ($184,500) | Salary Converter |
| A pension contribution taken before tax, and what it costs in take-home | UK salary sacrifice reduces the pay both income tax and National Insurance are charged on; a net pay arrangement reduces income tax only; adjusted net income, which drives the allowance taper above £100,000, falls with either. US 401(k) deferrals reduce federal and state taxable wages but not FICA; the 2026 deferral limit is $24,500 under 50 (ADP). salary_model.py works the £110,000 / £10,000 case by hand: £3,800 of take-home, 38p in the pound | Salary Converter |
| UK full new State Pension, £241.30 a week for 2026/27 | GOV.UK, new State Pension — 35 qualifying years for the full amount | Pension Pot Calculator |
| UK pension lump sum allowance £268,275 (a quarter tax-free), annual allowance £60,000, and the normal minimum pension age rising to 57 on 6 April 2028 | GOV.UK, lump sum allowance; annual allowance; normal minimum pension age | Pension Pot Calculator |
| US 2026 401(k) deferral limit $24,500, age 50 catch-up $8,000, age 60–63 catch-up $11,250 | IRS, 2026 retirement plan limits (Notice 2025-67) | Pension Pot Calculator |
| US Social Security: the share taxable at provisional income of $25,000 and $34,000 (single filers, not indexed) | IRS Publication 915 | Pension Pot Calculator |
| Canada 2026: RRSP dollar limit $33,810, and the 2026 federal brackets (14% to $58,523, then 20.5%, 26%, 29%, 33%) with a basic personal amount of $16,452 | Canada Revenue Agency, RRSP limits (read directly); CRA, income tax rates — the bracket thresholds and the basic personal amount were confirmed against several 2026 reports because that page could not be read directly | Pension Pot Calculator |
| Canada: CPP maximum $1,507.65 a month at 65 and an average new pension of about $877; Old Age Security about $762.50 a month at 65 to 74 | Service Canada, CPP amounts; OAS amounts — both are revised each quarter, so the page uses them as editable defaults | Pension Pot Calculator |
| Australia 2026-27: super guarantee 12%, concessional cap $32,500, maximum contribution base $270,830, and resident tax rates (15% from 1 July 2026) with 2% Medicare levy | ATO, key super rates and thresholds; ATO, resident tax rates — confirmed against several 2026-27 summaries because the ATO pages could not be read directly | Pension Pot Calculator |
| Pension pot projection and tax in retirement are this site’s own calculation | Month-by-month compounding with contribution limits and tax bands as above; checked in models/pension_model.py against hand arithmetic, the annuity formula and an independent implementation. Single-person, general-case figures, not personalised advice | Pension Pot Calculator |
| Debt avalanche vs. snowball payoff simulation | Standard amortisation mathematics — month-by-month interest accrual and payment allocation, verified against a hand-checked two-debt divergent case | Debt Payoff Calculator |
| Snowball method's behavioural completion-rate advantage | Gal, D. & McShane, B.B. (2012), "Can Small Victories Help Win the War? Evidence from Consumer Debt Management", Journal of Marketing Research 49(4) — closing accounts in ascending balance order predicted higher completion rates, despite avalanche minimising total interest | Debt Payoff Calculator |
| Rent-vs-buy opportunity-cost comparison methodology | Standard approach used by major rent-vs-buy calculators (e.g. New York Times) — net cost of buying vs. renting-and-investing the difference | Rent vs Buy Calculator |
| Claim | Source | Tool |
|---|---|---|
| Car-following model (Intelligent Driver Model) | Treiber, Hennecke & Helbing, "Congested traffic states in empirical observations and microscopic simulations", Phys. Rev. E 62, 1805 (2000) — acceleration exponent δ = 4 and the s* desired-gap form exactly as published. Parameter ranges offered here (T 0.4–3 s, a 0.3–4 m/s², b 0.5–5 m/s²) bracket the paper's own calibrated values. | Traffic Jam Simulator |
| Reaction-time delay, which standard IDM does not have | Added deliberately as an explicit delay on the perceived leader state, the usual formulation for delayed car-following. Plain IDM reacts instantly and is markedly more stable than human drivers; the page separates this from the desired time headway T, which is a distance rather than a lag, and explains why conflating them is the common error. | Traffic Jam Simulator |
| Phantom jams form on a ring road with no bottleneck | Sugiyama et al., "Traffic jams without bottlenecks — experimental evidence for the physical mechanism of the formation of a jam", New J. Phys. 10, 033001 (2008) — 22 vehicles on a 230 m circular track, which is the default scenario here. | Traffic Jam Simulator |
| Backward jam-wave speed ≈ (vehicle length + jam gap) ÷ start-up headway | Standard kinematic-wave result: the queue discharges at one vehicle per headway from a packing of one length plus gap, so the upstream front moves at their ratio. The page shows this prediction alongside a value measured from the simulation, so the two can be compared rather than asserted. | Traffic Jam Simulator |
| Flow = density × mean speed | The fundamental identity of traffic flow theory, used here to report vehicles per hour from the ring's own occupancy and average speed rather than by counting crossings. | Traffic Jam Simulator |
| PID control theory | Åström & Murray, Feedback Systems (Princeton, freely available) — standard modern reference | PID Control |
| Ziegler–Nichols, Cohen–Coon and lambda/SIMC tuning rules | Ziegler & Nichols reaction-curve rules (1942): Kp = 1.2τ/(Kθ), Ti = 2θ, Td = 0.5θ. Cohen & Coon (1953) for dead-time-heavy loops. Lambda/SIMC per Skogestad, with λ = 3θ as the robust default. All three as given in Åström & Murray, Feedback Systems and Seborg et al., Process Dynamics and Control. Each is re-derived in pid_model.py, which also asserts the identities they must all obey — doubling process gain halves Kp, more dead time backs every method off, and larger λ always detunes | PID Control |
| θ/τ as the measure of loop difficulty | The ratio of dead time to time constant is the standard controllability measure for FOPDT loops; above about 1 the loop is dead-time dominant and PID stops being the right tool, which is the same conclusion the site’s own “dead time beats PID” rule reaches. Bands used here: <0.1 easy, 0.1–1 normal, >1 dead-time dominant | PID Control |
| Nyquist-Shannon sampling theorem | Shannon, C.E. (1949), Proc. IRE 37(1), "Communication in the Presence of Noise" | Sampling & Aliasing |
| Quantisation: SNR = 6.02·N + 1.76 dB for a full-scale sine; triangular (TPDF) dither of ±1 LSB makes the error independent of the signal at a cost of 4.77 dB (three times the noise power), against 3.01 dB for rectangular dither | Lipshitz, S.P., Wannamaker, R.A. & Vanderkooy, J. (1992), “Quantization and dither: a theoretical survey”, Journal of the Audio Engineering Society — the 1.76 dB is 20·log10√(3/2); sampling_model.py derives both figures and measures the page’s own dither generator at 3.08× the undithered noise power | Sampling & Aliasing |
| Ball-and-beam double-integrator dynamics | Standard control-theory teaching example — see Åström & Murray, Feedback Systems, or any undergraduate controls course | Ball and Beam |
| Bode plot construction, gain/phase margin | Nyquist, H. (1932), Bell System Technical Journal; Bode, H.W. (1945), Network Analysis and Feedback Amplifier Design | Bode Plot |
| Claim | Source | Tool |
|---|---|---|
| The ladder itself: cache, memory, SSD, disk and network round trips | The figures every programmer has seen — Jeff Dean’s “latency numbers every programmer should know”, as maintained and updated by Colin Scott. They are order-of-magnitude values, not measurements, and the page says so: nothing is asserted to better than a factor of two, because no such figure exists for “an SSD” in general. What is exact is the arithmetic between the rungs. | Latency Ladder |
| Fan-out and tail latency: one slow backend in a hundred affects most requests | J. Dean & L. A. Barroso, “The Tail at Scale”, Communications of the ACM 56(2), 2013. If each of n backends independently has probability p of responding slowly, the chance that none does is (1−p)n — so at n = 100 and p = 1%, 63% of requests wait for a straggler. Independence is assumed and is the weak point: real slowness is correlated by shared hardware, shared garbage collection and shared load, which makes the true figure worse rather than better. The path builder sums its steps serially, which is the honest model for one thing after another and not for anything concurrent. | Latency Ladder |
| Memory to disk is 100,000×, which is five orders of magnitude | Definitional from the rungs: 100 ns against 10 ms. The Rulebook entry for this rule previously said six orders of magnitude and 100,000 in the same sentence, which cannot both be true; building this tool caught it and the entry now says five. | Latency Ladder |
| “Each rung roughly 100×” is wrong about one of its own two rungs | At the figures the rule itself states, memory to SSD is a factor of 1000 and SSD to disk a factor of 100. The mnemonic is a claim that the tiers are orders of magnitude apart rather than a constant ratio, and it understates the first gap — which is the one that matters most. The page shows both gaps against the 100× line. | Latency Ladder |
| One nanosecond as one second | An exact scaling: every figure is multiplied by 10⁹, so every ratio is preserved exactly. It is a presentation of the same numbers, not a different claim about them. | Latency Ladder |
| Effective throughput = size ÷ (access time + size ÷ sequential rate) | The standard access-plus-transfer model, and the reason the ladder stops applying to large reads. Sequential rates are order-of-magnitude figures for the media named. Queue depth is not modelled — a device serving many requests at once behaves very differently, and that difference is often larger than the difference between two media. | Latency Ladder |
| The rule’s own counterexample has expired | The entry notes that a spinning disk streaming sequentially can beat an SSD serving random 4 KB reads. Against a SATA SSD it still does, by about seven times. Against a modern NVMe drive it does not: roughly 205 MB/s of random 4 KB reads against 200 MB/s of streaming disk is a tie. The example was true when written and is now dated, which the page states rather than quietly dropping. | Latency Ladder |
| Losing every copy is the joint probability over four failure modes, not the product of four independent copies | Standard reliability practice: copies that share a room, a medium or a network are not independent, so they cannot be multiplied. The page enumerates every combination of media-type, site and logical (ransomware / propagated deletion) events exactly — no sampling — and lets the survivors of each combination fail independently within it. With the correlated modes switched off it collapses to the plain product, which is how the arithmetic is checked. | 3-2-1 Backup Calculator |
| Annualised drive failure rate ≈ 1.5% (default) | The only input on the page with a public dataset behind it: Backblaze publish quarterly drive statistics across a fleet in the hundreds of thousands, and the fleet-wide annualised failure rate has sat in roughly the 1.3–1.9% band for years. That is a datacentre with steady power and cooling; a drive in a drawer is worse by an amount nobody publishes. | 3-2-1 Backup Calculator |
| Media-type, site and logical event rates (defaults 0.5%, 0.2%, 2% a year) | Assumptions, not measurements, and the page says so beside each field. There is no published rate for "a firmware bug takes out a whole media type" because it is a category rather than a measurable population. The site figure is anchored loosely on dwelling fire statistics plus flood and theft. The logical figure varies by an order of magnitude depending on who is publishing it. All three are meant to be replaced by the reader. | 3-2-1 Backup Calculator |
| The probability that a surviving copy actually restores | No honest published figure exists. Every restore-failure rate in circulation comes from a company selling backup software, so the page presents this as a dial with a default rather than repeating any of them as though it were a measurement. Only the direction is agreed: a backup you have restored from is worth more than one you have not. | 3-2-1 Backup Calculator |
| Copies are never repaired or replaced during the horizon | Stated on the page. A drive that fails in year two is still failed in year twenty, so the risk shown is higher than a real setup where somebody notices and swaps it. The error runs towards pessimism, which is the direction a backup calculator should be wrong in. | 3-2-1 Backup Calculator |
| Allowed downtime = (1 − availability) × window | Definitional, and exact. A year is taken as 365 days and months as the length you select, because SLAs are written against calendar periods rather than the tropical year — which is why the same percentage buys fewer minutes in February than in March. | Uptime & SLA Calculator |
| Availability of services in series multiplies: A = ∏Aₕ | Standard reliability-block-diagram result for components in series, where the system needs every one of them. Three independent 99.9% dependencies give 99.7%, so a chain is always strictly worse than its weakest link. This is the rule's own stated failure mode — four nines on a three-nines dependency is unreachable at any price. | Uptime & SLA Calculator |
| n redundant copies: A = 1 − (1 − A)n | Standard parallel-block result, and it assumes statistically independent failures. Real replicas share regions, deployment pipelines, certificates and configuration pushes, so correlated failure is the normal case; the page presents this figure as a ceiling nothing reaches rather than a design target. | Uptime & SLA Calculator |
| An SLA is a service-credit threshold, not an engineering guarantee | Common structure across major cloud providers: falling below the published figure entitles the customer to a capped percentage credit against the next bill, usually on a calendar-month window and often only on request. The provider defines what counts as unavailable, and partial degradation frequently does not. The page states this rather than quoting any one provider's terms, which change. | Uptime & SLA Calculator |
| Binary prefixes: KiB = 210, MiB = 220, GiB = 230 bytes | IEC 80000-13 (2008, originally IEC 60027-2 amendment, 1998) — kibi, mebi, gibi. The standard settled the ambiguity; adoption never followed, so the same three letters still carry both meanings in ordinary use, which is what the page is actually about. | Bit Calculator |
| Any word width from 1 to 64 bits, with hex padded to ⌈bits/4⌉ digits and byte swap unavailable below a whole byte | Arithmetic on BigInt with an explicit width mask, so masking, two’s complement, rotation and population count are checked at every width from 1 to 64 in models/bitgrid_model.py rather than at the four that used to be offered | Bit Calculator |
| Lane striping deals bits round the lanes one at a time in reading order, so lane i receives display positions i, i+n, i+2n | The definition of bit-level interleaving. Verified by round trip in models/bitgrid_model.py: dealing the bits out and reading them back must return the original value, checked across every width, every lane count from 1 to 12 and both read directions — 1,800 combinations | Bit Calculator |
| Storage is sold in decimal units: 1 TB = 1012 bytes, which is 931.32 GiB | SI prefixes as used by every drive manufacturer, and the arithmetic is exact rather than a convention: 1012 ÷ 230 = 931.32. The shortfall compounds once per power — 2.34% at kilo, 11.18% at peta — which is why nobody disputes a kilobyte and everybody disputes a terabyte. Windows reports binary values under decimal labels; macOS and Linux report decimal. | Bit Calculator |
| TCP/IP over Ethernet keeps 94.93% of line rate at 1500 MTU (99.14% with 9000-byte jumbo frames) | Computed from the frame budget, not quoted: a 1500-byte IP packet carries 1460 bytes of TCP payload after the 20-byte IPv4 and 20-byte TCP headers, and occupies 1538 bytes on the wire once the 14-byte Ethernet header, 4-byte FCS, 8-byte preamble/SFD and 12-byte interframe gap are added — 1460 ÷ 1538. This is a wire-format ceiling only: it says nothing about wifi, contention, congestion control or the server at the far end, and the page says so. | Bit Calculator |
| Network rates are quoted in bits, file sizes in bytes — a factor of 8 | Long-standing convention: telecommunications counts bits per second, storage counts bytes. A 100 Mbit/s link therefore moves at most 12.5 MB/s. The page shows the mistaken reading alongside the correct one because it is the single most common error in the subject. | Bit Calculator |
| Probability an array survives its rebuild | Exponential (constant-hazard) failure model, p = 1 − e−AFR·t/8760 per surviving drive over the rebuild window, combined binomially with the unrecoverable-read-error term. Drive failures are treated as independent, which is the weakest assumption in the calculation — same model, same batch, same age, same workload, and a rebuild loads every survivor at once — so the result is a floor on the risk, and the page says so. The URE term uses the manufacturer's rated ceiling as though it were a measured per-bit rate, which is the pessimistic end of a long-running argument; real arrays rebuild successfully more often than it predicts. | RAID Calculator |
| RAID levels and fault-tolerance mathematics | Patterson, Gibson & Katz (1988), "A Case for Redundant Arrays of Inexpensive Disks (RAID)" — the original paper defining RAID | RAID Calculator |
| Prefix boundaries, block size ("magic number") and network overlap | Direct consequences of RFC 4632: a network address is the address with its host bits zeroed, so a prefix of length p can only begin on a multiple of 232−p, and two blocks overlap exactly when each begins before the other ends. Exact integer arithmetic with nothing to approximate; the overlap test is checked for symmetry and across the /31 and /32 edges in subnet_model.py. | Subnet Calculator |
| IPv4 subnetting (CIDR) | RFC 4632 — Classless Inter-Domain Routing | Subnet Calculator |
| Two's complement, IEEE-754-style bitwise semantics | Standard computer architecture reference — Patterson & Hennessy, Computer Organization and Design | Bit Calculator |
| Binary vs. decimal prefixes (KiB vs. kB) | IEC 60027-2 — the standard defining kibi-, mebi-, gibi- prefixes | Bit Calculator |
| Netflix 4K/HD minimum bandwidth (25 Mbps / 5 Mbps) | Netflix Help Center's own published internet speed recommendations | Internet Speed Test |
| Typing speed, filtering and percentile ranking | Words per minute uses the universal five-characters-per-word convention; net wpm subtracts one word per uncorrected error per minute, the convention used by the common online tests. Both are verified in typing_model.py, along with the three rolling filters and their published properties: an N-point moving average lags by (N−1)/2 samples, an exponential average of decay α has the same mean data age as a window of 2/α−1, and a rolling median is unmoved by a single outlier that shifts the mean by a third of it. Every one of those is asserted numerically rather than quoted. The percentile is from a lognormal distribution FITTED to two widely published anchors — a median near 40 wpm for adults and roughly the top 1% above 100 wpm — not from a survey and not from other users of this page; the page states this. Nothing typed here is transmitted or stored. Not modelled: keyboard layout, language, whether the text is copied or composed (copying is substantially faster), or the run-to-run variance of a short test, which typically exceeds the width of one band | Typing Speed Test |
| Typical movie file sizes (~15GB 4K, ~5GB 1080p, two-hour runtime) | Cross-referenced streaming/file-size references (GoBrolly, Dark Skies, Fortra) | Internet Speed Test |
| GTA V modern install size (~100GB, Enhanced edition with Online) | Cross-referenced across Rockstar/Steam listings and multiple gaming-press size reports | Internet Speed Test |
| Country-level median internet speeds worldwide | Ookla Speedtest Global Index, cross-referenced 2025-2026 snapshots — a representative sample, not a live feed | Internet Speed Test |
| Typical download:upload ratios by connection type | Cross-referenced ISP/connection-type technical references (fiber symmetric; cable/DSL asymmetric) | Internet Speed Test |
| Capacitor sizing Q = 2πfCV²; a delta bank needs one third the capacitance of a star bank for the same kVAR | Direct consequence of each capacitor seeing line voltage in delta and line/√3 in star, with reactive power scaling as V²; standard in power-factor correction practice and manufacturer bank design guides | Power Factor Calculator |
| Terminal velocity vt = √(2mg / ρCdA); free-fall time t = √(2h/g) in vacuum | Standard results from Newton's second law with quadratic drag (F = ½ρCdAv²); the vacuum case is Galileo's own, demonstrated on the Moon by Apollo 15 in 1971 | Galileo Drop Experiment |
| Equivalent dynamic load P = XFr + YFa; for single-row deep-groove ball bearings e and Y are functions of Fa/C₀ alone | ISO 281 rolling bearing dynamic load ratings; the e/Y table is reproduced identically in the SKF, NSK, NTN and FAG general catalogues, confirming it is type-dependent rather than part-dependent | Bearing Calculator |
| Kirchhoff's current law (ΣI = 0 at a node) and voltage law (ΣV = 0 around a loop) | Gustav Kirchhoff, 1845 — consequences of conservation of charge and of energy respectively; standard in every circuit-analysis text | Electrical Circuit |
Password entropy is length × log₂(alphabet) for random strings and words × log₂(list) for passphrases; a 7,776-word Diceware list gives 12.9 bits per word against 6.6 for a character from a 95-symbol set | Shannon entropy applied to a uniform generator. Diceware list size is Reinhold’s original 6⁵ = 7,776. NIST SP 800-63B withdrew composition rules and mandatory rotation, and recommends length over character-class requirements, for the reasons the tool demonstrates | Password Strength & Crack Time |
Sample size per variant for two proportions is (zα/2+zβ)²[p₁(1−p₁)+p₂(1−p₂)]/(p₂−p₁)², so halving the detectable effect roughly quadruples the sample | Standard two-sample proportion power calculation with a normal approximation to the binomial; the inverse normal uses Acklam’s rational approximation. The 5% baseline / 10% relative lift case returns ~31,000 per arm, matching published sample-size calculators | A/B Test Sample Size Calculator |
| Peeking at a test of no effect: false-positive rate after K equally spaced looks (8.3% at 2, 14.2% at 5, 19.3% at 10, 24.8% at 20, at a 5% threshold); several variants judged at the Šidák level; a finished test read with a Newcombe interval on the difference | Armitage, P., McPherson, C.K. & Rowe, B.C. (1969), “Repeated significance tests on accumulating data”, Journal of the Royal Statistical Society A — the page computes the same probability for any threshold, and abtest_model.py checks the two-look case by direct integration. Šidák, Z. (1967), Journal of the American Statistical Association. Newcombe, R.G. (1998), “Interval estimation for the difference between independent proportions”, Statistics in Medicine | A/B Test Sample Size Calculator |
| Converting money between two years is the ratio of consumer price index values, and £100 in 1970 is about £1,609 in 2025 money | UK CPI (ONS, 2015 = 100), annual averages to 2025 plus the latest published month, with pre-1988 values from the ONS CPI-consistent modelled historical series and pre-1950 values from the ONS long-run composite price index, chained onto CPI at 1950 so that nothing from 1950 onward is affected; US CPI-U (BLS, 1982–84 = 100). Recent UK years are derived from the published annual-average rates (1.8, 0.9, 2.6, 9.1, 7.3, 2.5%), and the resulting series reproduces the Bank of England calculator's landmarks: £1 in 1970 → £16.09, in 1990 → £2.61, in 2000 → £2.00. The US series reproduces the BLS figure of $100 (1980) → $380.70 and $1 (1913) → $31.69 | Inflation Calculator |
| Rest between sets of roughly 3–5 minutes for strength, 60–90 seconds for hypertrophy and 30–60 seconds for muscular endurance; one-rep-max estimated from a submaximal set; barbell plate loading and session volume load | Rest bands follow standard strength-and-conditioning guidance (ACSM and NSCA position stands on resistance training). One-rep max uses three published regressions shown side by side rather than one: Epley (1985), Brzycki (1993) and Lombardi (1989). Their divergence is computed rather than asserted — within ~5 kg at five reps on a 100 kg lift, 76.8 kg at twenty; Epley and Brzycki cross exactly at ten reps, and Epley reads 3.3% high at a true single because it is not calibrated there. Working-weight percentages are the inverse of Epley. Plate loading is a greedy per-side fill that never overshoots. Tailoring adds BMI (mass over height squared, with the standard 18.5/25/30 bands), reference body weight from Devine (1974) and Robinson (1983) — both defined in inches over five feet, which is why they read as imperial — and starting loads seeded from strength standards expressed as multiples of bodyweight, taken at the middle of the commonly published training-standard ranges and used only as a starting point. Session length is time under tension plus the rests between sets. All re-derived in gym_model.py (82 checks, including an exhaustive sweep of every half-kilo from 20–200 kg). General guidance for healthy adults, not medical advice | Gym Companion |
| Waist under half your height is a single boundary that works for both sexes and from childhood, and catches cases BMI misses in both directions | Waist-to-height ratio with a 0.50 boundary, as used by NICE guidance on assessing central adiposity; the two disagreement cases are worked directly — a 180 cm 95 kg athlete is BMI 29.3 with a ratio of 0.46, a 180 cm 78 kg sedentary person is BMI 24.1 with a ratio of 0.53. Lower cut-offs apply for some ethnic groups | Waist-to-Height Ratio Calculator |
| Induction motors draw 6–8× full-load current at standstill, and that band straddles the trip window of an ordinary breaker | Full-load current from P/(√3·V·PF·η) with the plate rating as shaft output; locked-rotor multiples from motor nameplate code letters. Magnetic trip windows of 3–5×, 5–10× and 10–20× for type B, C and D are the standard IEC 60898 curves. A 7.5 kW 400 V motor computes to 14.2 A running, 85 A at 6× and 113 A at 8× against a 20 A type C window of 100–200 A | Motor Starting & Inrush |
| Reduced-voltage starting cuts current in proportion to voltage and torque with the square of it, so star-delta’s symmetric one-third is a coincidence of the connection | Induction motor torque is proportional to the square of applied voltage at fixed slip, current to the first power. The delta reconnection divides line current and torque by exactly three; an autotransformer tap a divides both by a²; a solid-state starter divides current by a and torque by a². A drive holds volts-per-hertz constant and so is not governed by the square law at all | Motor Starting & Inrush |
| A PI controller driving a saturating actuator overshoots substantially, and anti-windup removes it without changing a single gain | Simulated directly: a first-order plant (τ = 10 s, unity gain) under a parallel-form PI controller (Kc = 2, Ti = 8 s) stepped to 10 with the output clipped at 12 overshoots 12.8% unguarded and 0.0% with conditional integration. Back-calculation follows Åström and Hägglund’s formulation. With the clip raised to 200 all three behaviours agree to within half a point, confirming the effect is saturation rather than tuning | Integral Windup & Anti-Windup |
| Dead time widens the swing beyond the deadband and puts a floor under it | First-order-plus-dead-time, the standard process-control model (Seborg, Edgar & Mellichamp, Process Dynamics and Control). Switching is delayed by θ, so the peak is C − (C − hi)e−θ/τ and the trough amb + (lo − amb)e−θ/τ — both fixed points immediately, with nothing to iterate. The overshoot is therefore proportional to the distance the plant is driving past the limit, so an oversized unit suffers most; and the residual swing at zero deadband, (C − amb)(1 − e−θ/τ), is a floor no thermostat setting can reach. Modelled as a single pure delay, which lumps sensor lag, sensor placement and plant response together. Verified in onoff_model.py. | On-Off Control & Hysteresis |
| On-off cycling rate peaks near 50% duty, and halving the deadband roughly doubles the number of cycles | Classical bang-bang result for a first-order plant: on-time and off-time are the exponential crossing times of the deadband, and because the heating ceiling is ambient plus the unit’s authority, both extremes of load send one of the two to infinity. Manufacturer limits of roughly six starts per hour and ten minutes minimum run are typical across compressor and boiler documentation | On-Off Control & Hysteresis |
| A first-order-plus-dead-time step response reaches 63.2% of its final value one time constant after the delay ends, and 98.2% after four; open-loop tuning constants follow from K, τ and L alone | 63.2% is 1 − e−1 by definition. Tuning relations are the published open-loop rules: Ziegler & Nichols (1942) reaction curve, Cohen & Coon (1953), and lambda / internal model control tuning as given in Seborg, Edgar & Mellichamp | First-Order Lag & Dead Time |
| Temperature derating, cycle life against depth of discharge, and cost per kWh delivered | Capacity is rated at 25 °C and falls below it: lead-acid by roughly 0.8% per degree, lithium chemistries by nearer 0.3%, because the loss is ionic mobility in the electrolyte rather than a slowing reaction. Held at 100% above 25 °C rather than extrapolated upward — a hot battery is a shorter-lived one, not a bigger one. Cycle life uses the standard manufacturer curve shape, life = N·(ref/DoD)w, fitted at the depth each chemistry is normally rated at; w > 1 for lead-acid is why halving the depth of discharge roughly triples the life while costing only half the energy per cycle, so for lead-acid the cost per kilowatt-hour delivered falls as the bank is worked less hard, while for LiFePO₄ (w = 0.85) it does not. Life is capped at a typical calendar life of about 8, 12 and 10 years for lead-acid, LiFePO₄ and NMC (design-life ranges from manufacturer datasheets, not a measurement of any one battery), and the at-your-loads figures divide the bank’s cycles by the cycles a day that your loads use, counting the inverter’s idle draw for all 24 hours. Lithium must not be charged below freezing (lithium plating, permanent and cumulative), which is a safety limit rather than a performance one. Prices are typical UK retail and will not be yours; the comparison between chemistries is the useful output, not the absolute pence. Verified in battery_model.py. | Battery Sizing & Runtime |
Delivered capacity falls with discharge current per Peukert’s law t = H(C/IH)k, with k ≈ 1.1–1.3 for lead-acid and ≈ 1.05 for LiFePO₄; usable depth of discharge is about 50% and 80–90% respectively | Peukert (1897), still the standard high-rate model; exponent and depth-of-discharge ranges are the values consistently given across lead-acid and LiFePO₄ manufacturer datasheets. The formula returns the rated time exactly at the rated current for any exponent, which is the internal check that it is being applied correctly | Battery Sizing & Runtime |
| Sodium lost in sweat, and the hyponatraemia bound on drinking | Sweat sodium spans roughly 200–2,000 mg/L between individuals — a five-fold range, far wider than sweat volume varies, and not inferable from anything a scale measures, which is why the page makes it a setting and not a constant. Heat acclimatisation over about two weeks roughly halves it while raising sweat volume. Sodium chloride is 39.34% sodium by mass, so 1 g of sodium is 2.542 g of salt; both are shown because labels and guidelines quote either without always saying which. The upper bound on drinking is the important half: exercise-associated hyponatraemia follows from replacing salty sweat with plain water, and has killed runners who finished heavier than they started — hence "never plan to drink more than you sweat". Losses of 1–2% of body mass across a session are normal. This page models volume and a sodium concentration you choose; only a patch test measures yours. | Sweat Rate & Hydration |
| Sweat rate is body mass change, plus fluid taken in, minus urine passed, divided by duration, taking 1 kg as 1 L; losing about 2% of body mass measurably impairs endurance performance, and rates range roughly 0.5–2.5 L/h between individuals | American College of Sports Medicine position stand on exercise and fluid replacement; the mass-balance method and the 2% threshold are standard in it. Guidance on drinking to thirst rather than ahead of it follows the exercise-associated hyponatraemia consensus statements, which reversed earlier advice after fatalities in endurance events | Sweat Rate & Hydration |
| Maximum heart rate falls with age at roughly 0.7 beats/year (208 − 0.7 × age); the older 220 − age overestimates in the young and underestimates in the old, crossing it at age 40. Zones by heart rate reserve exceed the same percentage of maximum by resting × (1 − percentage) | Tanaka, Monahan & Seals (2001), meta-analysis of 351 studies / 18,712 subjects; Fox (1971) for 220 − age; Nes et al. (2013) HUNT study; Gulati et al. (2010) for the women-derived equation. The reserve identity follows directly from Karvonen’s definition | Heart Rate Zones & Max HR |
| Training zones from a measured lactate threshold heart rate instead of an estimated maximum | Joe Friel’s field test and running zones: the average heart rate over the last 20 minutes of a 30-minute solo effort as LTHR, with zone 1 under 85%, zone 2 85–89%, zone 3 90–94%, zone 4 95–99% and zone 5 from 100% (his 5a–5c). hr_model.py pins the boundaries and checks that no maximum-formula uncertainty enters them | Heart Rate Zones |
| DC charging holds near peak power to a knee around 50–60% state of charge then tapers, so the last 20% can take as long as the first 70%; charging losses are roughly 12% on home AC and 6% on DC | Constant-current / constant-voltage lithium-ion charging — taper is required to hold cell voltage inside a safe window as the battery fills. Loss figures are consistent across published metered charging tests; AC is worse because rectification happens in the onboard charger | EV Charging & Running Cost |
| ISO week 1 is the week containing the year's first Thursday (equivalently 4 January); weeks run Monday–Sunday and an ISO year has 53 weeks when 1 January is a Thursday, or a Wednesday in a leap year | ISO 8601-1:2019 §4.2.2, date and time representations for information interchange — the 53-week condition follows directly from the week-1 rule rather than being separately defined | Week Number Calculator |
| All three hands together, and how often each pair meets | Direct consequences of the hand speeds (hour 30°/h, minute 360°/h, second 360°/min): a pair meets once per lap the faster hand gains on the slower, so hour–minute meet 11 times in twelve hours, minute–second 708 (59 an hour, not 60) and hour–second 719 — each one less than the naive answer, because the hand being chased is also moving. All three coincide only at 12:00:00: the hour–minute overlaps fall at multiples of 12/11 h, whose fractional parts put the second hand elsewhere at every one but the last. The closest approach away from noon — a spread of 0.50° at 8:43:43.65 and at its mirror image 3:16:16.35 — is a searched result at hundredth-of-a-second resolution, not a closed form; the twin follows because every hand angle negates under t → 12 h − t, and the model finds it by its own search too. Counts verified both ways in clock_model.py, by closed form and by exhaustive search. | Clock Angle Calculator |
| Clock hand angle = |30H − 5.5M|; hands coincide every 720/11 minutes (11 times in 12 hours) | Direct consequence of the hand rates — minute hand 6°/min, hour hand 0.5°/min — giving a 5.5°/min closing speed; standard result in recreational and competition mathematics | Clock Angle Calculator |
| 60 pixels per degree is the resolving limit of 20/20 vision; thresholds scale with the Snellen ratio | Snellen definition of visual acuity (20/20 resolves 1 arcminute; 60 arcminutes per degree), consistent with the ~30 cycles/degree contrast-sensitivity limit of the healthy eye | Screen Size & Viewing Distance |
| Recommended horizontal field of view 30° (SMPTE) to 40° (THX) | SMPTE EG-18 cinema viewing recommendation and THX screen placement guidance | Screen Size & Viewing Distance |
| Transfer time = size / throughput (Mbps vs MB/s factor of 8) | Definitional relationship between bit-rate and byte-rate; same estimates as Internet Speed Test use-cards | Data Transfer Time Calculator · also Speed Test |
| Ethernet protocol overhead is 5.07% at a 1500-byte MTU and 0.86% with jumbo frames, so a gigabit link carries at most 949 Mbps of payload | Frame arithmetic, not an estimate: 20 B IP + 20 B TCP headers inside the frame, and 14 B Ethernet header + 4 B FCS + 8 B preamble + 12 B inter-frame gap outside it — 1460 B of payload per 1538 B on the wire, and 8960 B per 9038 B at MTU 9000 | Data Transfer Time Calculator |
| TCP throughput is capped at window ÷ round-trip time, so a 64 KB window over a 10 ms link cannot exceed 52 Mbps however fast the line is | A sender may have only one window in flight per round trip; the ceiling is the same statement as the bandwidth-delay product, and parallel streams multiply it because each carries its own window. Verified against the identity that a window equal to the BDP exactly fills the link | Data Transfer Time Calculator |
| Typical sustained sequential storage speeds by device class — USB 2.0 35, SD card 90, 7200 rpm hard disk 120, USB 3.0 400, SATA SSD 550, NVMe 3500 MB/s | Representative figures for each class of device, not measured for any specific product. Used only to establish which ceiling binds: a 7200 rpm disk at 960 Mbps is just above the ~930 Mbps a gigabit link delivers as payload, so it does not bind, while an SD card at 720 Mbps does | Data Transfer Time Calculator |
| Claim | Source | Tool |
|---|---|---|
| Greenwich mean sidereal time; altitude and azimuth from the hour angle | The standard GMST polynomial (IAU 1982/2006 convention, as given in the Explanatory Supplement to the Astronomical Almanac), anchored to 18h 41m 50.55s at the J2000 epoch. The alt/az conversion is the ordinary spherical-triangle result, checked against facts that hold independently of any catalogue: the altitude of the pole equals the latitude; a star on the celestial equator rises due east from every latitude; a star transits when local sidereal time equals its right ascension. | Star Map & Constellations |
| Positions of 104 bright stars, J2000 | Hand-entered right ascensions, declinations and visual magnitudes at equinox J2000.0, of the kind published in the Bright Star Catalogue and every standard star atlas. A catalogue cannot be derived, so it is verified by geometry: 24 angular separations that anyone can look up — Orion's belt end to end, the three sides of the Summer Triangle, the Pointers to Polaris, both axes of the Southern Cross — are computed from the shipped page and checked in stars_model.py. No precession, nutation, aberration, parallax, proper motion or refraction, so a position can be out by up to about half a degree. | Star Map & Constellations |
| c, h and the electronvolt as exact values; hc = 1239.84 eV·nm | SI base unit definitions (2019 redefinition) — the speed of light, the Planck constant and the elementary charge are all defined exactly, so hc is exact arithmetic rather than a measured quantity. This is what makes the 1240 rule stable. | Electromagnetic Spectrum Explorer |
| Atmospheric transmission windows (optical, thermal IR 8–13 µm, radio) | A schematic, and labelled as one on the chart itself: a piecewise-linear model in log wavelength through representative clear-sky, sea-level, zenith values, with the principal absorbers (O3 below ~300 nm, H2O at 1.4, 1.9, 2.7 and 6.3 µm, CO2 at 4.3 and 15 µm, O2 at 60 GHz, H2O at 22 GHz, ionospheric cut-off near 10 MHz) placed where they occur. It is not a line-by-line radiative-transfer calculation; window edges and band positions are real, percentages are indicative and move with humidity, altitude and airmass. | Electromagnetic Spectrum Explorer |
| Wien displacement constant b = 2.897771955×10−3 m·K | CODATA / NIST, from the transcendental root of the Planck peak condition. Boltzmann's constant is itself exact since 2019. | Electromagnetic Spectrum Explorer |
| Visible light rendered in true colour from wavelength | CIE 1931 2° colour matching functions, via the multi-lobe piecewise-Gaussian fits of Wyman, Sloan & Shirley, Simple Analytic Approximations to the CIE XYZ Color Matching Functions, JCGT 2(2), 2013 — then XYZ to sRGB (D65) with out-of-gamut colours desaturated towards white rather than clipped per channel, which would shift the hue. The strip is what an average eye reports for a single pure wavelength, not a photograph. | Electromagnetic Spectrum Explorer |
| Planck spectral radiance, and the fraction of a blackbody's power that is visible | Planck's law, B(λ,T) = (2hc²/λ⁵)/(ehc/λkT−1), evaluated with expm1 for numerical stability at short wavelengths; the visible fraction is a Simpson integration over log λ of the same function, so numerator and denominator share their discretisation error. | Electromagnetic Spectrum Explorer |
| Reference energies: kT at room temperature, hydrogen bond ~0.2 eV, C–C bond ~3.6 eV, water ionisation 12.6 eV | Standard physical chemistry values. The ionisation figure is the vertical ionisation energy of the water molecule; the page presents it as a threshold that depends on the target rather than a universal cliff, because the widely quoted "10 eV" boundary for ionising radiation is a convention, not a constant. | Electromagnetic Spectrum Explorer |
| Band edges (radio / microwave / IR / visible / UV / X-ray / gamma) | Common conventions, which differ between fields; the page says so rather than presenting them as physics. X-rays and gamma rays genuinely overlap in wavelength and are distinguished by ORIGIN — electron transitions versus nuclear — not by any boundary on the axis. | Electromagnetic Spectrum Explorer |
| The field of a long straight wire, B = μ0I/2πr, the force per metre between two wires, μ0I1I2/2πd, and the force on a current, BIL sin θ, with μ0 = 1.25663706212×10−6 N/A² | NIST CODATA 2018 for the vacuum permeability; Ampère’s law and the Biot-Savart law (Jackson, Classical Electrodynamics, 3rd edition, chapter 5). The model sums Ampère’s law numerically round circles that hold one wire, two wires and two opposite wires, and gets μ0 times the enclosed current each time. | Magnetism |
| The field and the flux function of a circular current loop in complete elliptic integrals K and E, from which the solenoid, the Helmholtz pair and the bar magnet (a sheet of current Br/μ0 round its side) are built | Smythe, Static and Dynamic Electricity (3rd edition, chapter 7) and Jackson (3rd edition, section 5.5, the vector potential of a loop). The model sums the Biot-Savart law round the ring numerically at 65 points (agreement to 2×10−10 of the field), integrates the field over a disc to check the flux function, checks that the divergence and the curl of the field vanish, and compares the solenoid and the magnet with their exact closed forms on the axis. | Magnetism |
| The Helmholtz pair (two coils one radius apart) has the field (4/5)3/2 μ0NI/R = 0.7155 μ0NI/R at the centre, with a field flat to fourth order in the distance from it | Hermann von Helmholtz’s arrangement, set out in any text on electromagnetism from the on-axis field of a loop; the model derives the coefficient and the fourth-order flatness, B(z)/B(0) − 1 = −(144/125)(z/R)4, from the field of two loops. | Magnetism |
| A charged particle of kinetic energy T in a uniform field B goes round a circle of radius p sin θ/(|q|B) at angular frequency |q|B/(γm), with the relativistic momentum pc = √(T(T + 2mc2)); the masses of the ions are the atomic masses less the electron’s | NIST CODATA 2018 for the elementary charge and the masses of the electron, proton and atomic mass unit; the Lorentz force and cyclotron motion. The model integrates the relativistic equation of motion in three dimensions with scipy for an electron, a proton and an alpha particle at several energies and angles, and compares the radius, the period, the pitch and the path (agreement to a few parts in a million of the radius or better); the same integration checks the mass spectrometer. | Magnetism |
| The flux through a coil from a moving magnet, the voltage ε = −N dΦ/dt (Faraday’s law and Lenz’s law), and the generator peak voltage NBAω | Faraday’s law of induction (Jackson, chapter 5 and section 5.15). The model integrates the field over the disc of the coil for the flux, shows that a coil small enough to see a uniform field gives −Nπa2 dB/dt to 0.2 percent, that the voltage integrates to nothing over a whole pass, and compares the generator with a numerically differentiated flux. | Magnetism |
| The Earth’s main field is the World Magnetic Model 2025: the spherical-harmonic coefficients to degree 12 with their yearly change, valid from 2025.0 to 2030.0, in the geodetic frame of the WGS 84 ellipsoid; it predicts the main field to about half a degree of declination and 100 nT | NOAA National Centers for Environmental Information and the British Geological Survey, the World Magnetic Model, with the coefficient file WMM2025.COF and the published test values. The model compares the 540 coefficients on the page with the file exactly, reproduces all twelve test points (X, Y, Z, H, F, inclination, declination) to the 0.1 nT and 0.01° they are given to, and checks the field at 200 random points against a second implementation written with scipy’s Legendre functions and a numerical gradient of the potential, to 0.001 nT. | Magnetism |
| The outline of the land on the map of declination (the 110 m coastline, simplified to 8.6 kB), which is not used for any number | Natural Earth, 1:110m physical vectors, land, in the public domain. | Magnetism |
| Damped harmonic oscillator equations | Standard 2nd-order ODE result — Halliday & Resnick, Fundamentals of Physics, ch. on oscillations | Mass-Spring-Damper |
| Multi-link pendulum equations of motion (RK4 integration) | Lagrangian mechanics, standard derivation — e.g. Taylor, J.R., Classical Mechanics | Pendulum Simulator |
| Coulomb friction model (static/kinetic) | Standard introductory mechanics result; friction coefficients from Marks' Standard Handbook for Mechanical Engineers | Inclined Plane |
| A rolling body accelerates at g·sinθ/(1+k) where k = I/mr², so a hoop gets exactly half a frictionless slide and a solid sphere five sevenths — independent of mass and radius | Standard rigid-body result from combining Newton's second law along the slope with the torque equation about the contact point; k is 2/5 for a solid sphere, 1/2 for a solid cylinder, 2/3 for a hollow sphere and 1 for a hoop. Verified: mass and radius cancel algebraically, and the ordering sphere < cylinder < hollow sphere < hoop holds at every angle | Inclined Plane |
| Rolling without slipping requires μ ≥ tanθ × k/(1+k) — 0.17 for a sphere at 30°, 0.29 for a hoop | The friction force needed to supply the angular acceleration, divided by the normal force. Below it the body slips and reverts to the sliding case, arriving sooner because it no longer spends energy on rotation | Inclined Plane |
| Pushing a load up a ramp needs mg(sinθ + μcosθ), giving a mechanical advantage of 1/sinθ at best and an efficiency of sinθ/(sinθ + μcosθ) | Force balance along the slope at constant velocity. Verified that with μ = 0 the work done equals the lifting work exactly, so a ramp trades force for distance and never saves energy; a 1:12 ramp at μ 0.2 is 29% efficient | Inclined Plane |
| The angle of repose is the critical angle, so μ = tan(repose): dry sand 34°, dry soil 35°, gravel 45° | Typical published repose angles for granular materials; the identity with the critical angle is definitional, since a heap stands at exactly the angle where the surface grains are on the point of sliding | Inclined Plane |
| Newton's law of cooling; convective/radiative heat transfer | Incropera & DeWitt, Fundamentals of Heat and Mass Transfer | Beverage Cooling Simulator |
| The specific heat of water is 4,184 J/(kg K); warming liquid water from 0 to 100 °C takes 418 kJ per kilogram, boiling it 2,256 kJ, and warming steam by 20 K about 41 kJ | NIST Chemistry WebBook, thermophysical properties of fluid systems (the IAPWS-95 formulation), the enthalpy of water at one atmosphere, compared in models/heat_model.py at fifteen temperatures. The energy to warm liquid water agrees to about 0.5 percent (the specific heat is 4.22 at 0 °C and 4.18 near 40 °C, and the tool takes one value), and the latent heat of vaporisation to 0.1. | Specific Heat & Heat Transfer |
| The latent heat of fusion of ice is 334 kJ/kg and the specific heat of ice is about 2,050 J/(kg K) | IAPWS and standard tables (333.55 kJ/kg at 0 °C); the specific heat of ice at −10 °C from the standard table of specific heat capacities. | Specific Heat & Heat Transfer |
| The specific heats and thermal conductivities of the materials in the table: typical values near 20 °C, in which real samples of wood, concrete, brick and insulation vary by 10 to 30 percent and the specific heat rises with temperature | The standard tables of specific heat capacity and of thermal conductivity (CRC Handbook of Chemistry and Physics, as collected in Wikipedia’s tables of specific heat capacities and thermal conductivities); the building materials are typical design values in the range of ISO 10456. Each is checked in the model against the published value or range. | Specific Heat & Heat Transfer |
| Melting points, boiling points and latent heats of fusion and vaporisation of ethanol, aluminium, copper, iron, lead and mercury | CRC Handbook molar enthalpies (vaporisation and fusion) divided by molar mass; the tool’s values agree to within 2.5 percent. Iron’s and aluminium’s vaporisation values differ by about 3 percent between editions. The same specific heat is used for the solid, liquid and gas of these (only water has three), which is an estimate across a change of state. | Specific Heat & Heat Transfer |
| The final temperature of a mixture is the mean of the temperatures weighted by mass times specific heat; ice in water is settled by adding enthalpy, with liquid water at 0 °C as zero and ice at 0 °C as −Lf per kilogram | Conservation of energy (calorimetry). The model checks the mixture on 40 random cases for energy conservation and the ice case against three hand-worked branches (all the ice melts, some melts, the water freezes). | Specific Heat & Heat Transfer |
| The U-value is one over the sum of the layer resistances d/k and the surface films (0.13 and 0.04 m²K/W); an unventilated air gap is 0.18; single glazing comes to about 5.7 W/(m²K) and ordinary double glazing about 2.8 | ISO 6946, building components, thermal resistance and thermal transmittance, horizontal heat flow. The model solves the same wall as a network of conductances with a linear solver and compares every face temperature. | Specific Heat & Heat Transfer |
| The dew point from the Magnus formula with a = 17.62 and b = 243.12 °C | Sonntag (1990); the model compares it with the dew point from Buck’s (1996) saturation vapour pressure, written separately, at six temperatures and humidities, within 0.12 K. | Specific Heat & Heat Transfer |
| Radiation from a surface is εσ(Ts4 − Tr4) with σ = 5.670374419×10−8 W/(m²K4); a surface at room temperature acts as a convection coefficient of 4εσT3, about 5.7 W/(m²K) for a black one | NIST CODATA (the Stefan-Boltzmann constant is exact in the SI). The black-body emission at 300 K, 459.3 W/m², is checked in the model. | Specific Heat & Heat Transfer |
| Convection coefficients: about 6 W/(m²K) in still air, 5.7 + 3.8v in a wind of v m/s (17 at 3 m/s, 44 at 10), 400 for still water and 3,000 for flowing water; uncertain by a third | Typical ranges for free and forced convection of gases and liquids (Incropera, DeWitt et al., Fundamentals of Heat and Mass Transfer, table 1.1) and the McAdams (1954) correlation 5.7 + 3.8v for a smooth surface. | Specific Heat & Heat Transfer |
| A body can be treated as one temperature when the Biot number hL/k, with L the volume over the surface area, is under about 0.1; its cooling then follows mc dT/dt = −A[h(T − Ta) + εσ(T4 − Tr4)] | The lumped-capacitance method (Incropera et al., chapter 5). The tool integrates it by fourth-order Runge-Kutta; the model compares the time to reach a temperature with scipy’s adaptive integrator at six cases to 0.01 percent, and with the exact exponential when the radiation is off. | Specific Heat & Heat Transfer |
| 3-iron, 5-iron and gap-wedge launch figures | Interpolated within the same PGA Tour Trackman ladder as the other clubs in this tool — each sits between its neighbours on ball speed, launch angle and spin. Not separately measured averages; they complete a standard 14-club bag for comparison purposes | Projectile Motion & Golf Ball Flight |
| Driver ball-speed distributions by standard of player | Approximate cohort means and standard deviations drawn from published launch-monitor datasets — Trackman tour averages for the tour row, and club-fitting / Arccos amateur distributions for the rest. Rounded deliberately: they describe populations, not individuals, and should not be read as a ranking | Projectile Motion & Golf Ball Flight |
| Whole-bag estimation from a single calibrated club | Method, not a citation: ball speed is scaled by the reader's ratio to the tour figure for their selected club, and spin at 55% of that ratio. It assumes bag ratios are consistent within a player, which holds far better than absolute numbers but breaks down for a genuinely uneven bag — stated in the tool's field notes | Projectile Motion & Golf Ball Flight |
| Quadratic drag equation, drag coefficients | Hoerner, S.F., Fluid-Dynamic Drag — standard reference for drag coefficient by shape | Galileo Drop Experiment, Projectile Motion |
| Apollo 15 hammer-and-feather drop | NASA Apollo 15 mission transcript/video — Commander David Scott, 2 August 1971 | Galileo Drop Experiment |
| Golf ball flight — Magnus effect, launch conditions | TrackMan/Titleist Performance Institute published launch-condition data; USGA equipment rules for legal driver COR | Projectile Motion & Golf |
| PGA Tour average driving distance | PGA Tour official statistics | Projectile Motion & Golf |
| Kepler's laws, vis-viva equation | Kepler, J. (1609/1619), Astronomia Nova / Harmonices Mundi; modern treatment in Curtis, H.D., Orbital Mechanics for Engineering Students | Orbital Mechanics |
| Planetary orbits, sizes, masses, temperatures and compositions | Orbital elements are the standard J2000 osculating values (semi-major axis, eccentricity, inclination, longitude of ascending node, longitude of perihelion, mean longitude); radii are IAU mean radii and masses are derived from published GM values. Only those are typed. Orbital periods, mean and extreme orbital speeds, surface gravities, escape velocities, bulk densities, equilibrium temperatures, insolation, synodic periods and every Earth-relative ratio are computed in solar_model.py from G, the solar mass and those elements, and checked against published figures before emission — derived periods land within 0.6% of published values, worst case. Positions come from solving Kepler’s equation numerically. Equilibrium temperature is the bare-body radiative balance and is stated as a rule of thumb that fails: it misses Venus by over 500 °C, which the page shows rather than hides. Escape velocities and gravities are computed at the MEAN radius, so they differ slightly from the commonly published equatorial-radius figures (Jupiter 60.2 km/s here against 59.5); the page says so. Not modelled: mutual perturbations, so no Mercury perihelion advance and no way to derive Neptune from Uranus; orbital precession; moon orbits; anything requiring an ephemeris | Solar System |
| Time dilation, length contraction, GPS corrections and relativistic journeys | Special relativity in flat spacetime, plus weak-field gravitational time dilation. The Lorentz factor is derived in relativity_model.py from the light-clock triangle rather than quoted, and a check confirms Pythagoras reproduces it at every speed tested. Two external anchors are reproduced from first principles: the NIST 2010 optical-clock experiment, which measured a 33 cm height difference as roughly 90 ns over 79 years (model: 89.9 ns), and the published GPS corrections — −7.2 µs/day from orbital speed, +45.8 µs/day from altitude, net +38.6 µs/day and about 11.6 km of position drift per day uncorrected. Muon survival, relativistic velocity addition, rest energy and Schwarzschild radii are all checked against published values. The journey calculator uses constant proper acceleration with a mid-point turnover, which is honest mechanics and wildly optimistic engineering. Not modelled: spacetime curvature, tensors, frame dragging, anything at or inside a black hole horizon, relativistic aberration or brightness. Diagrams are schematic and are drawn to be understood rather than to scale | Relativity |
| Central-body radii, and which radius an altitude is measured from | Earth uses the equatorial radius, 6378.137 km (WGS 84), because altitude on this page is r − R and that is the convention every published orbital altitude follows. It is not a rounding preference: the page previously used the mean radius, 6371 km, and so placed geostationary orbit at 35,793 km against the universally quoted 35,786 km — seven kilometres out, in the one figure on the page a reader is most likely to already know. The other bodies keep their standard mean radii (Moon 1737.4, Mars 3389.5, Jupiter 69,911, Sun 695,700 km); no altitude above those is a number anyone checks against, and their oblateness is not modelled here either way. μ values are the JPL/IAU GM constants. orbital_model.py now reads both μ and the radius back out of this page and asserts they match its own, and separately that the geostationary altitude the page would display is 35,786.03 km | Orbital Mechanics |
| ISS orbital altitude, period, velocity | NASA ISS Facts and Figures — ~400 km altitude, ~90–93 min period | Orbital Mechanics |
| Standard gravitational parameters (μ) by body | NASA JPL Solar System Dynamics | Orbital Mechanics, Rocket Equation |
| Hohmann transfer Δv and transfer time between circular orbits | Standard two-impulse result from vis-viva: Δv₁ = √(μ/r₁)(√(2r₂/(r₁+r₂))−1), Δv₂ = √(μ/r₂)(1−√(2r₁/(r₁+r₂))), transfer time π√(a³/μ) for a = (r₁+r₂)/2. Curtis, Orbital Mechanics for Engineering Students; Vallado, Fundamentals of Astrodynamics and Applications. orbital_model.py reproduces the published LEO–GEO figures (2.455 + 1.477 = 3.932 km/s over 5.26 h) and checks that the transfer ellipse’s periapsis and apoapsis speeds match the post- and pre-burn speeds to machine precision | Orbital Mechanics |
| Bi-elliptic transfer becomes cheaper past a radius ratio of about 11.94 | Classic three-impulse result; the 11.938765… crossover for an infinitely distant intermediate apoapsis is given in Curtis and Vallado. orbital_model.py locates it numerically by bisection and agrees to 5×10⁻³. It is the standard counter-example to “Hohmann is the minimum-energy transfer”, which is stated more confidently than it deserves | Orbital Mechanics |
| Tsiolkovsky rocket equation | Tsiolkovsky, K.E. (1903), "The Exploration of Cosmic Space by Means of Reaction Devices" | Rocket Equation & Delta-v Budget |
| Falcon 9 / Saturn V / Electron vehicle specifications | SpaceX, NASA (Saturn V Flight Manual SA-503), Rocket Lab public specifications — approximate figures assembled for illustration, not manufacturer data sheets | Rocket Equation & Delta-v Budget |
| Typical mission delta-v budgets (LEO, GTO, TLI, escape) | Curtis, H.D., Orbital Mechanics for Engineering Students — typical, trajectory-dependent published estimates | Rocket Equation & Delta-v Budget |
| 1D/2D coefficient-of-restitution collision equations | Standard classical mechanics — any introductory physics or engineering dynamics textbook (e.g. Halliday & Resnick, Fundamentals of Physics) | Collision Simulator |
| Newton's Cradle "N in, N out" behaviour from equal-mass elastic collisions | Standard result of solving simultaneous momentum and kinetic-energy conservation for equal masses — verified numerically in this tool against the textbook signature | Collision Simulator |
| Claim | Source | Tool |
|---|---|---|
| RYB subtractive colour mixing model | Gossett, N. & Chen, B. (2004), "Paint Inspired Color Mixing and Compositing for Visualization" — the RYB trilinear interpolation approach this tool is based on | Paint Colour Mixing |
| Refresh rate measured from animation-frame callbacks | W3C HTML Living Standard — requestAnimationFrame runs once per presented frame. There is no browser API that reports a display’s refresh rate, so this is an inference from presentation timing, not a hardware reading | Screen Refresh Rate Test |
| Why the estimate uses a median rather than an average | Measured, not cited: refresh_model.py runs the estimator against synthetic frame streams of known rate. A 240 Hz stream with 3% dropped frames averages out to about 232 Hz; the median-and-filter estimate returns 240 and reports the drops separately | Screen Refresh Rate Test |
| Timestamp resolution, and why results are aggregates | Browsers deliberately quantise performance.now() to mitigate timing side-channel attacks (Spectre); the page measures and displays the resolution it actually gets. At 240 Hz on a 1 ms clock every individual interval reads 4 or 5 ms and never 4.17 | Screen Refresh Rate Test |
| The list of standard refresh rates, and the snapping tolerance | Conventional panel rates. The 1.2% tolerance is set by the closest pair in that list (175/180 Hz, 2.78% apart) so no suggestion is ambiguous — a reading further out is reported as measured rather than rounded to a rate the panel may not have | Screen Refresh Rate Test |
| Snell's law, the critical angle, and total internal reflection | Standard geometric optics: n₁ sinθ₁ = n₂ sinθ₂, with θc = arcsin(n₂/n₁) | Refraction & Diffraction |
| Reflectance of a transparent boundary, and Brewster's angle | The Fresnel equations in their real-valued dielectric form, for s- and p-polarised light; Brewster’s angle arctan(n₂/n₁) is where the p-component reaches zero. Not valid for metals or absorbing media, which need a complex refractive index | Refraction & Diffraction |
| Diffraction angles for slits, gratings and circular apertures | Fraunhofer (far-field) diffraction: d sinθ = mλ, read as minima for a single slit of width d and as maxima for a spacing d; the circular aperture uses the 1.22 λ/D Airy/Rayleigh criterion | Refraction & Diffraction |
| Refractive indices of common materials | Conventional textbook values at the sodium D line (589 nm). Real materials vary by formulation and with wavelength — the page states this rather than implying a measurement | Refraction & Diffraction |
| Light transport in the room preview: bounced light, soft shadows and colour bleeding between surfaces | Path tracing after Kajiya, J. (1986), The Rendering Equation, with explicit light sampling (next-event estimation) for the window, sun and ceiling fitting | Room Colour Visualiser |
| How sheen changes a surface, from matt to gloss | GGX/Trowbridge-Reitz microfacet distribution with Smith masking and a Schlick Fresnel term at F0 = 0.04, the standard dielectric value. The roughness assigned to each sheen name is illustrative: paint is sold by gloss units measured at a fixed angle and manufacturers do not agree on where one name stops and the next begins | Room Colour Visualiser |
| Light reflectance value (LRV) quoted for a wall colour | CIE Y tristimulus value under D65, expressed as a percentage — the same quantity the trade quotes on a colour card | Room Colour Visualiser |
| Tone mapping of the rendered image | The ACES filmic approximation (Narkowicz, 2015). Needed because a sunlit wall beside a window spans a far wider range of brightness than a screen can show | Room Colour Visualiser |
| The fourteen tube colours, and their positions in the mixing model | Representative masstone renderings of each pigment, with the RYB coordinates fitted to reproduce them under the model above rather than typed by hand — subject to a cap on each tube’s implicit black content, without which a tube fitted to a dark pigment darkens every mix it joins. Not measured spectral data for any particular manufacturer’s paint, which is what accurate prediction would need | Paint Colour Mixing |
| sRGB transfer function and primaries (hex/RGB conversions, the linear segment near black) | IEC 61966-2-1:1999, Default RGB colour space — sRGB | Paint Colour Mixing |
| CIE L*a*b* under the D65 white point, and ΔE*ab (CIE76) as the colour-difference measure | CIE 15:2004, Colorimetry — CIE76 is a plain Euclidean distance in Lab and is known to overstate differences among saturated blues, which is why the page presents ΔE as a ranking | Paint Colour Mixing |
| Contrast ratios quoted against white and black | W3C WCAG 2.2, relative luminance and contrast ratio definitions | Paint Colour Mixing |
| How much chroma a mix can reach at each hue (the gamut readout and hue chart) | Measured, not cited: colour_model.py takes the maximum chroma in each Lab hue bucket for the mixable set and for the sRGB gamut surface. It is a property of the RYB model above, not of any real paint range — the model places its red and yellow primaries on the sRGB primaries, so the warm end reads as fully reachable by construction | Paint Colour Mixing |
| Coats needed, from the colour being covered | Each coat hides the same fraction of what is still showing, so the surviving fraction after n coats is (1−H)n — the standard contrast-ratio model behind every manufacturer's opacity claim. Films are mixed in linear light and compared in CIE L*a*b* using CIE76 ΔE, with the conventional thresholds: ΔE < 1 imperceptible, < 2 perceptible only side by side. Hiding power is an input rather than a property of a named paint, because published contrast ratios are measured under conditions nobody reproduces on a wall. ΔE76 rather than ΔE2000: for same-hue pairs a short distance apart they agree closely, and ΔE2000 would imply a precision the hiding-power input does not have. Sheen and texture are not modelled at all. Verified in paintcoverage_model.py. | Paint Coverage Calculator |
| Paint coverage rates by type (emulsion, masonry, gloss, primer) | Compiled from multiple UK trade decorating references — illustrative, genuinely varies by product and manufacturer | Paint Coverage Calculator |
| Porosity and wastage adjustment factors for paint calculation | UK trade decorating guidance — new plaster/porous surfaces reduce coverage; 10% minimum wastage allowance The porosity factor is applied to the first coat only, since the first coat is the one a porous surface soaks up; every later coat is at the tin’s own coverage. Cost is for whole 2.5 L tins at the price per litre entered, so it is an upper figure: bigger tins are usually cheaper per litre | Paint Coverage Calculator |
| UK stud spacing standards (400mm/600mm centres) | UK trade joinery and drylining references, tied to plasterboard thickness (9.5mm → 400mm, 12.5mm → 600mm) | Screw & Stud Sizing Selector |
| Plasterboard fixing practical load ceiling (~20kg) | Independent destructive testing of real fixings, finding plasterboard itself fails around 60kg of axial load regardless of fixing type — a third of peak load is the trade-standard safety margin | Screw & Stud Sizing Selector |
| Wall plug diameter bands by load | UK fixing manufacturer and retailer guidance (Toolstation, ManoMano) | Screw & Stud Sizing Selector |
| The rule of twelfths is within 2.6% of the range of a true sinusoid, and exact at the 2, 3 and 4 hour marks | Derived here rather than quoted: the rule is a standard piece of navigational practice (RYA and Admiralty teaching), and the error figures are computed directly against a half-cosine of the same range and duration. The worst case, 2.56% of range, sits about 0.62 h after high water | Tide Curve & Rule of Twelfths |
| A semidiurnal tide runs 6h 12m from high water to low, and sea level moves about 1 cm per millibar of pressure | The lunar day is 24h 50m, so a semidiurnal cycle is 12h 25m and each half 6h 12m. The inverse barometer effect is the standard approximation of roughly 1 cm of sea level per hPa away from 1013 — enough that a deep depression raises the water further than the rule of twelfths is ever wrong by, and it is in no tide table | Tide Curve & Rule of Twelfths |
| Radiated energy is log10 E = 1.5 M + 4.8 joules, so each magnitude step is 31.6 times the energy and 10 times the amplitude; moment magnitude is Mw = ⅔(log10 M0 − 9.1) | Gutenberg and Richter (1956) for the energy relation, an empirical average that scatters by about a factor of three; Hanks and Kanamori (1979) for moment magnitude, with M0 in newton-metres. TNT equivalents use 4.184×109 J a tonne, Hiroshima is taken as 15 kt (6.3×1013 J), and a typical UK home as 2,700 kWh a year. Equivalent power divides the energy by a rupture time of length over 2.8 km/s, about 80% of the shear-wave speed | Earthquake Magnitude & Richter Scale |
| Rupture length, area and average slip are log10 L = −2.44 + 0.59 M, log10 A = −3.49 + 0.91 M and log10 D = −4.80 + 0.69 M | Wells and Coppersmith (1994), all slip types combined, subsurface rupture length. They were fitted to magnitudes 4.8 to 8.1 on crustal faults and are extrapolated above that, where subduction megathrusts run longer than the formula says | Earthquake Magnitude & Richter Scale |
| The map shades each 2° cell by its count of magnitude 6 and larger earthquakes from 1973 to 2025; the Gutenberg–Richter slope is b = 1.00 between M6 and M7, and the S−P gap is 8.4 km a second | The USGS ComCat catalogue, magnitude 6 and above, fetched once, with the 70 explosions removed (7,326 earthquakes from 1973). It starts in 1973 because earlier catalogues are patchy in remote regions and would shade the places that had seismographs; fifty-three years is also short against the centuries between the great quakes of slow faults such as Cascadia, which the map therefore shows as quiet. The slope is log10(7,326 / 729) from the catalogue’s own counts at M6 and M7. Above M6 the page uses the measured counts, which fall away faster than a straight line (40 quakes of M8 and over, where the line gives 73); below M6 it extends the line, which lands within about 5% of the USGS long-run estimates (about 1.45 million a year of M2 and over). Plate boundaries are the PB2002 model (Bird, 2003) simplified to 0.25°. The gap rule uses VP = 6.0 and VS = 3.5 km/s, and magnitude from the gap and amplitude is the Hutton and Boore (1987) Southern California correction, valid 10 to 600 km | Earthquake Magnitude & Richter Scale |
| Death tolls for the named earthquakes, and the Haiti and Anchorage comparison | Commonly quoted figures from the USGS event pages and national agencies, rounded. Estimates vary, sometimes by a factor of two or more: Haiti 2010 runs from about 100,000 to over 300,000, Tangshan 1976 from the official 242,000 to far higher, and the 2023 Turkey–Syria total is quoted as over 55,000. Magnitudes and positions are the catalogue’s where it holds the event; the 2011 Fiji quake at 558 km depth is from it too | Earthquake Magnitude & Richter Scale |
| Ascent rates: 30 ft/min (9.1 m/min) for the US Navy and NOAA, 10 m/min for Bühlmann tables and many dive computers, 15 m/min for BSAC 88 and MN90 tables, 60 ft/min (18 m/min) as the older agency maximum | DAN (Divers Alert Network), “Ascent Rates”: 30 ft/min is the current US Navy and NOAA standard, the Navy having changed from 60 about twenty years before 2012; agency recommendations range 30–60 ft/min; a 2009 study found higher bubble grades at 60 than at 30 ft/min. The Bühlmann 10 m/min, the 1–5 minute safety stop at 3–6 m and variable-limit computers are from the decompression-practice literature (Wikipedia, “Decompression practice”); the Suunto 10 m/min limit is from the maker’s manual; the BSAC 88 and MN90 15 m/min and the 7–20 m/min depth-varying range of one maker are as reported in diver-community sources, and are the least well sourced figures on the page. Rates change with each edition: check your own agency, table or computer manual. The page does not compute no-stop limits or decompression | Dive Depth & Ascent Guide |
| Oxygen partial pressure limits of 1.4 bar (working) and 1.6 bar (contingency), and the maximum operating depth MOD = 10 × (ppO2 / FO2 − 1) | The limits commonly taught for nitrox and technical diving, and the arithmetic of partial pressure (Dalton’s law) with the diver’s convention of 10 m of seawater to one bar; the physical figure is about 9.95 m and the Altitude & Depth tool uses it. Air at 1.4 bar is 56.7 m, which the model recomputes | Dive Depth & Ascent Guide |
| Depth records and bands: deepest scuba 332.35 m, deepest breath-hold (no limits) 214 m, deepest open-sea saturation 534 m, atmospheric diving suit 610 m, Alvin 6,500 m, Challenger Deep 10,935 m; Trieste’s 1960 ascent took 3 h 15 min | Guinness World Records for the scuba dive (Ahmed Gabr, 18 September 2014, Red Sea), the no-limits breath-hold dive (Herbert Nitsch, 14 June 2007) and the Challenger Deep depth of 10,935 m ± 6 m (Victor Vescovo, DSV Limiting Factor, 2020); COMEX Hydra 8 (1988) for the saturation record; the US Navy dive of August 2006 for the atmospheric diving suit; the Woods Hole Oceanographic Institution for Alvin’s 6,500 m and its 30 m/min; the US Navy history account of the 1960 Trieste dive. The coloured depth bands are an editorial grouping of the depths agencies and manuals use (18, 30 and 40 m recreational limits), not a standard | Dive Depth & Ascent Guide |
| Wait at least 12 hours before flying after a single no-stop dive, 18 hours after several dives or several days, and substantially longer after decompression dives; 24 hours as a simple rule | DAN, “Guidelines for Flying After Diving”, for divers without symptoms and for cabin altitudes of 2,000–8,000 ft. Other agencies, computers and doctors may give longer figures | Dive Depth & Ascent Guide |
| Mean adult height and BMI of each country, by sex | NCD Risk Factor Collaboration (NCD-RisC), pooled population-based surveys: height of adults born in 1996 (the 2016 eLife “century of trends in adult height” series) and age-standardised adult BMI to 2016 (Lancet, 2017), as published by Our World in Data. These are country means: the spread is not in the source and is modelled on the page as 4% of the mean for height and 0.16 in ln BMI, typical of national survey data. Weight is derived from BMI and height; foot length as 15% of height with 0.8 cm of scatter; both are modelled and graded as such | Where You Stand |
| Income per person per day in 2021 international dollars, mean and median by country, 2023 | The World Bank Poverty and Inequality Platform, per person per day in 2021 purchasing-power-parity dollars, 214 economies, from household surveys of income or of consumption. Each country’s distribution is a lognormal set by its mean and median. Checked against the Platform’s own country headcounts, weighted by the same populations: the fit puts 12.9% of the world under $3 a day where the surveys give 10.5%, so the poorest tail is overstated by about 2.4 points, most of it in India (10% against 4%) and Indonesia. The page converts a local-currency figure with the World Bank’s 2021 PPP factor for private consumption and the consumer price index since 2021 | Where You Stand |
| Net worth: one world curve, about 41% of adults under $10,000 and about $90,000 mean wealth per adult | UBS Global Wealth Report (2025 and 2026 editions): the share of adults under $10,000 was “just over 41%” in 2025 and total household wealth about $471 trillion in the 2025 edition. The page fits a lognormal to those two facts, giving a median near $15,000 and a spread (sigma of the log) of 1.88. It is a single world curve because no open per-country table of adult wealth was found; the lognormal understates the very rich, and a net worth of zero or less is outside it | Where You Stand |
| Age, and the national averages of carbon, electricity, water, meat and schooling | Population by broad age band, 2023, from the UN World Population Prospects via Our World in Data (spread evenly inside each band, so coarse); CO₂ per person (Global Carbon Project, fossil, production-based), electricity generated per person (Ember and the Energy Institute), freshwater withdrawals per person (FAO AQUASTAT), meat supply per person (FAO) and mean years of schooling (Barro–Lee), the latest year available to 2023. These are national averages per person, so they are ranked against other countries’ averages weighted by population, never against individuals | Where You Stand |
| Resting heart rate: 70 beats a minute for men and 74 for women, with a spread of 10 | A single world estimate from the published adult cohort literature (adult means of roughly 70–75 with a standard deviation near 10); there is no open country dataset, so it is used without country detail and graded rough | Where You Stand |
| Port tidal ranges shown are nominal typical figures, not predictions | Rounded representative mean spring and neap ranges, included for scale and for the contrast between ports — the spread from Avonmouth to Poole is the most surprising thing about British tides. They are NOT a tide table and must not be used for navigation; take real heights and times from an almanac | Tide Curve & Rule of Twelfths |
| Hyperfocal distance H = f²/(Nc) + f, with near and far limits u(H−f)/(H+u−2f) and u(H−f)/(H−u) | The standard thin-lens depth-of-field geometry, used in its exact form rather than the usual small-angle simplifications. The circle of confusion c is a criterion for acceptable blur, not a constant: the traditional value is the sensor diagonal over 1500, and every depth-of-field figure anywhere depends on that choice | Depth of Field |
| The speed of sound in an ideal gas is √(γRT/M), independent of pressure | Standard gas dynamics: for air γ=1.4 and M=28.9645 g/mol, giving 340.29 m/s at 15 °C, the International Standard Atmosphere sea-level value. Real air departs from the ideal-gas figure by well under 0.1% here; humidity adds up to about 0.5% and is not modelled | Speed of Sound & Mach Cone |
| Thin-airfoil lift slope of 2π per radian, the Prandtl–Glauert factor 1/√(1−M²) and Ackeret’s supersonic lift 4α/√(M²−1) | Anderson, J.D., Fundamentals of Aerodynamics (McGraw-Hill), chapters 4, 11 and 12; Abbott, I.H. and von Doenhoff, A.E., Theory of Wing Sections (Dover) | Wind Tunnel |
| Oblique-shock and normal-shock relations used to check the shock solver, including the wedge and wall-reflection cases | NACA Report 1135, Equations, Tables, and Charts for Compressible Flow (Ames Research Staff, 1953) | Wind Tunnel |
| Sod shock tube, the exact test of a compressible-flow solver | Sod, G.A. (1978), Journal of Computational Physics 27, 1–31 | Wind Tunnel |
| The lattice Boltzmann method (D2Q9) and its Smagorinsky eddy-viscosity extension | Qian, d’Humières and Lallemand (1992), Europhysics Letters 17; Hou, Succi, Chen and Sterling (1996), Journal of Computational Physics 118 | Wind Tunnel |
| The HLLC approximate Riemann solver and MUSCL reconstruction used for the high-speed flow | Toro, E.F., Spruce, M. and Speares, W. (1994), Shock Waves 4, 25–34; Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer) | Wind Tunnel |
| Steady flow past a cylinder at Reynolds number 20, drag coefficient about 2.05, used as the low-speed solver’s benchmark | Tritton, D.J. (1959), Journal of Fluid Mechanics 6, 547–567; Dennis, S.C.R. and Chang, G.-Z. (1970), Journal of Fluid Mechanics 42, 471–489 | Wind Tunnel |
| Typical stall angles of real wings (about 12 to 18 degrees) and that slats, flaps and spoilers act as described | Anderson, Introduction to Flight and Fundamentals of Aerodynamics; Abbott and von Doenhoff, Theory of Wing Sections | Wind Tunnel |
| A boundary layer separates where the speed at the wall falls to zero, and beyond that point the air next to the surface runs backwards; the layer thickens with distance and thins with Reynolds number | Schlichting, H. and Gersten, K., Boundary-Layer Theory (Springer), chapters 2 and 7; Prandtl (1904) | Wind Tunnel |
| The wing, slat, flap and spoiler shapes, the grids and the immersed-boundary method are this site’s own | Checked in models/windtunnel_model.py. A two-dimensional teaching simulation at low Reynolds number: not a design tool | Wind Tunnel |
| Planck’s law, with h, c and k at their exact SI values, and the Stefan–Boltzmann constant 5.670374419×10⁻⁸ W/m²K⁴ that follows from them | BIPM, The International System of Units (SI Brochure), 9th edition (2019); Tiesinga, E. et al. (2021), CODATA recommended values 2018, Reviews of Modern Physics 93, 025010 | Black Body & Colour Temperature |
| Wien’s displacement constant, 2.897771955 mm·K, and that the peak of the curve per unit wavelength is at 2,898 micrometre-kelvins divided by the temperature | Tiesinga et al. (2021), CODATA 2018 | Black Body & Colour Temperature |
| The CIE 1931 colour-matching functions, and the sum-of-lobes fit to them used for the colour of the glow | CIE 15:2004, Colorimetry; Wyman, C., Sloan, P.-P. and Shirley, P. (2013), Journal of Computer Graphics Techniques 2(2), 1–11 | Black Body & Colour Temperature |
| The Planckian locus (the chromaticity of a black body) and the published table of black-body colours used to check the swatches | Kim, Y.S. et al. (2002), Journal of the Korean Physical Society 41, 865–871; Charity, M., “What color is a blackbody?” (vendian.org/mncharity/dir3/blackbody) | Black Body & Colour Temperature |
| Conversion from CIE XYZ to a screen colour (linear sRGB with a D65 white, then the sRGB transfer curve) | IEC 61966-2-1:1999, Colour management: default RGB colour space, sRGB | Black Body & Colour Temperature |
| The Sun’s nominal radius (6.957×10⁸ m), effective temperature (5,772 K) and luminosity (3.828×10²⁶ W) | IAU 2015 Resolution B3; Prša, A. et al. (2016), The Astronomical Journal 152, 41 | Black Body & Colour Temperature |
| The luminous efficacy of 683 lm/W at 555 nm, and that a black body’s radiation gives at most about 95 lm/W, near 6,600 K | BIPM, SI Brochure (the candela); Ohno, Y. (2004), Color rendering and luminous efficacy of white LED spectra, Proc. SPIE 5530, 88–98 | Black Body & Colour Temperature |
| The Draper point: solids begin to glow dull red at about 525 °C (798 K) | Draper, J.W. (1847), Philosophical Magazine 30, 345–360 | Black Body & Colour Temperature |
| Lamp colour temperatures (about 2,700 K warm white to 6,500 K daylight), and the approximate surface temperatures and radii of Proxima Centauri, Sirius A, Betelgeuse and Rigel | ANSI C78.377 (chromaticity of solid-state lighting); Cox, A.N. (ed.), Allen’s Astrophysical Quantities (Springer). Stellar values are approximate and differ by source | Black Body & Colour Temperature |
| Emissivities: about 0.95 for skin, paint and soot, about 0.05 for polished metal | Incropera, F.P. et al., Fundamentals of Heat and Mass Transfer (Wiley), appendix tables | Black Body & Colour Temperature |
| The colours, spectra and efficacies on the page are this site’s own calculation | Checked in models/blackbody_model.py. A perfect black body on a screen: not for lighting design or for measuring a temperature | Black Body & Colour Temperature |
| Avogadro’s constant, 6.02214076×10²³ per mole (exact), used with the mass number as the molar mass to turn grams into a number of atoms | BIPM, The International System of Units (SI Brochure), 9th edition (2019) | Radioactive Decay & Half-Life |
| The half-lives and main decay modes of the sixteen nuclides (carbon-14, tritium, cobalt-60, caesium-137, strontium-90, iodine-131, technetium-99m, fluorine-18, nitrogen-16, polonium-212, radon-222, americium-241, radium-226, potassium-40, plutonium-239, uranium-238), and what each leaves | Kondev, F.G. et al. (2021), The NUBASE2020 evaluation of nuclear properties, Chinese Physics C 45, 030001; Evaluated Nuclear Structure Data File (National Nuclear Data Center, Brookhaven) | Radioactive Decay & Half-Life |
| The half-life of carbon-14 as 5,730 years, rather than the 5,568 of the original radiocarbon convention or the 5,700 ± 30 evaluated since | Godwin, H. (1962), Half-life of radiocarbon, Nature 195, 984 | Radioactive Decay & Half-Life |
| The year as 365.25 days, for converting half-lives to seconds | IAU Resolution B2 (1976); SI Brochure (the Julian year) | Radioactive Decay & Half-Life |
| The law of radioactive decay: the exponential, the decay constant, the mean life and activity | Rutherford, E. and Soddy, F. (1902), Philosophical Magazine 4, 370–396; Krane, K.S., Introductory Nuclear Physics (Wiley, 1988), chapter 6 | Radioactive Decay & Half-Life |
| Radiocarbon dating, and that it reaches back to about 50,000 years | Libby, W.F., Radiocarbon Dating (University of Chicago Press, 1955); Reimer, P.J. et al. (2020), IntCal20, Radiocarbon 62, 725–757 | Radioactive Decay & Half-Life |
| That alpha particles are stopped by paper, beta particles by a few millimetres of aluminium, and gamma rays only weakened by thick lead | Knoll, G.F., Radiation Detection and Measurement, 4th edition (Wiley, 2010); IAEA Safety Glossary (2018) | Radioactive Decay & Half-Life |
| The becquerel as one decay per second, and the curie as 3.7×10¹⁰ of them | BIPM, SI Brochure (9th edition, 2019) | Radioactive Decay & Half-Life |
| The proportion of potassium that is potassium-40, 0.0117% | Meija, J. et al. (2016), Isotopic compositions of the elements 2013, Pure and Applied Chemistry 88, 293–306 | Radioactive Decay & Half-Life |
| About 7,000 decays a second in a person (potassium-40 about 4,400, carbon-14 about 3,000), and about 15 in a banana | UNSCEAR (2008), Sources and Effects of Ionizing Radiation, Annex B; Health Physics Society, Radiation Fact Sheets | Radioactive Decay & Half-Life |
| A smoke alarm holding about 33 kBq of americium-241, and a diagnostic dose of technetium-99m of about 600 MBq | US Nuclear Regulatory Commission, Backgrounder on Smoke Detectors; ARSAC, Notes for Guidance on the Clinical Administration of Radioactive Substances (Public Health England) | Radioactive Decay & Half-Life |
| The simulation, the activities of a gram of each nuclide and the reference scale are this site’s own calculation | Checked in models/radioactivity_model.py. A simple model of one decay step: not for planning work with radioactive material | Radioactive Decay & Half-Life |
| The principle of superposition, standing waves on a string fixed at both ends, and the harmonics n v / 2L | French, A.P., Vibrations and Waves (Nelson, 1971), chapters 3 and 5; Feynman, R.P., Leighton, R.B. and Sands, M., The Feynman Lectures on Physics, volume I, chapters 48 and 49 | Standing Waves & Superposition |
| Pipes open at both ends and closed at one end: the modes, the odd harmonics of the closed pipe and the open-end correction of about 0.6 of the radius | Fletcher, N.H. and Rossing, T.D., The Physics of Musical Instruments, 2nd edition (Springer, 1998), chapter 8; Levine, H. and Schwinger, J. (1948), Physical Review 73, 383–406 | Standing Waves & Superposition |
| The steady response of a damped string or air column to a force at a point (a Green function of the wave equation with a complex wave number) | Morse, P.M. and Ingard, K.U., Theoretical Acoustics (McGraw-Hill, 1968), chapter 9 | Standing Waves & Superposition |
| The speed of sound in air, 343 m/s at 20 °C, rising about 0.6 m/s for each degree | Cramer, O. (1993), Journal of the Acoustical Society of America 93, 2510–2516 | Standing Waves & Superposition |
| Note names and frequencies: A4 = 440 Hz in equal temperament, 100 cents to a semitone | ISO 16:1975, Acoustics — Standard tuning frequency | Standing Waves & Superposition |
| The instrument presets: a guitar’s 648 mm scale and low E (82.4 Hz), a violin’s 325 mm string and A (440 Hz), a flute near C4, an organ pipe near C2, a clarinet-like pipe near E3 | Wolfe, J., Music Acoustics (University of New South Wales, newt.phys.unsw.edu.au/music); Fletcher and Rossing (1998), chapters 9, 15 and 16 | Standing Waves & Superposition |
| A microwave oven’s 2.45 GHz, a wavelength of 12.2 cm and a hot-spot spacing of 6.1 cm | ITU Radio Regulations, article 5.150 (the industrial, scientific and medical band at 2,450 MHz) | Standing Waves & Superposition |
| Beats at the difference of the two frequencies, and the envelope of two added waves | Feynman, Leighton and Sands (1963), volume I, chapter 48 | Standing Waves & Superposition |
| The simulation, the resonance curve and the pictured air are this site’s own calculation | Checked in models/standingwaves_model.py. Ideal waves in one dimension: not for designing an instrument | Standing Waves & Superposition |
| The thin lens and mirror equation, 1/u + 1/v = 1/f, the sign convention (real is positive), the magnification −v/u, and the three principal rays | Hecht, E., Optics, 5th edition (Pearson, 2017), chapter 5; Young, H.D. and Freedman, R.A., University Physics (Pearson), chapter on geometric optics | Lenses & Mirrors |
| The lens maker’s equation, 1/f = (n − 1)(1/R1 − 1/R2), and a mirror’s focal length R/2 | Hecht (2017), sections 5.2 and 5.4 | Lenses & Mirrors |
| That a spherical mirror brings rays far from the axis to a focus nearer the mirror (spherical aberration), by R/(2 cos θ), and that a parabola has none for parallel rays | Smith, W.J., Modern Optical Engineering, 4th edition (McGraw-Hill, 2008), chapter 6; Hecht (2017), chapter 6 | Lenses & Mirrors |
| Power in dioptres as 1/f with f in metres, and the addition of the powers of thin lenses in contact | ISO 13666:2019, Ophthalmic optics — Spectacle lenses — Vocabulary; Hecht (2017), section 6.1 | Lenses & Mirrors |
| The near point of the eye, 25 cm, and the magnifying power of a simple magnifier, 25/f for a relaxed eye and 25/f + 1 with the image at the near point | Hecht (2017), section 5.7; Young and Freedman, geometric optics | Lenses & Mirrors |
| The angular diameter of the Sun, 32 arcminutes (0.5333°), and so an image of it f/107 across | NASA Goddard Space Flight Center, Sun Fact Sheet (apparent diameter 31.6–32.7 arcminutes) | Lenses & Mirrors |
| That wing mirrors of the commonest classes are convex with a radius of curvature of not less than 1.2 m | UN Economic Commission for Europe, Regulation No. 46, Devices for indirect vision (classes II and III) | Lenses & Mirrors |
| Camera focusing by moving the lens: the lens moves f²/(u − f) from its position for infinity | Hecht (2017), section 5.2 | Lenses & Mirrors |
| The image positions, the fan of rays, the blur on a screen and the spherical-aberration picture are this site’s own calculation | Checked in models/lensesmirrors_model.py. Thin lenses and mirrors near the axis: not for designing an optical system | Lenses & Mirrors |
| Temperature with height: 15 °C falling 6.5 °C/km to −56.5 °C at 11 km, constant to 20 km, then warming 1 °C/km | The International Standard Atmosphere (ICAO Doc 7488; identical to the US Standard Atmosphere 1976 below 32 km). The page's day-temperature control shifts the whole profile, as ISA+/− figures are used in aviation | Speed of Sound & Mach Cone |
| The Mach cone half-angle is arcsin(1/M), and a boom reaches the ground only where the aircraft outruns the sound at every height below it (Mach cutoff) | Classical supersonic flow and acoustic ray theory in a horizontally stratified, still atmosphere: the shock's local angle satisfies sin μ = a(z)/v, so it lags the aircraft by ∫√(M(z)²−1) dz and cannot exist where M(z)≤1. This is the basis of the "Mach cutoff" flight studied by NASA for quiet supersonic flight over land. Wind and inversions are not modelled | Speed of Sound & Mach Cone |
| Boom Supersonic's XB-1 flew supersonic with no boom recorded on the ground (Mach 1.12 on 28 January 2025; Mach 1.18 at about 36,500 ft on 10 February 2025), marketed as “Boomless Cruise” | Boom Supersonic, “Boomless Cruise” announcement and February 2025 flight update (boomsupersonic.com). The company's own report; the Mach 1.3 figure for Overture is its stated goal, not a demonstrated result | Speed of Sound & Mach Cone |
| NASA's X-59 is shaped so its boom reaches the ground as a soft thump; first flight October 2025, first supersonic flight June 2026 | NASA Quesst mission pages and flight updates (nasa.gov/mission/quesst). Community overflight results were not yet published when this was written | Speed of Sound & Mach Cone |
| Civil supersonic flight over US land has been banned since 1973 (14 CFR 91.817); a 2025 executive order directed the FAA to replace it with a noise-based standard, proposed in July 2026 | Executive Order 14304, “Leading the World in Supersonic Flight”, Federal Register 11 June 2025; FAA proposed rule “Enabling Supersonic Overland Flight”, Federal Register 2 July 2026. A proposal, not a final rule, as of September 2026 | Speed of Sound & Mach Cone |
| Diffraction blur is the Airy disc at 2.44λN, reaching the circle of confusion near f/21 on full frame | The Airy disc diameter to the first null is 2.44λN; taken at λ=550 nm, green light at the eye's peak sensitivity. Setting it equal to the d/1500 circle of confusion gives the crossover aperture per format. Lens aberrations, which improve on stopping down, are not modelled, so a real lens usually peaks a little tighter than this suggests | Depth of Field |
| Easter falls on the first Sunday after the ecclesiastical full moon on or after 21 March, computable for any year | The anonymous Gregorian computus, the algorithm in use since the 1582 reform. It uses a TABULAR full moon rather than an astronomical one, so it can differ from the real full moon by a day or two — that is the rule, not an error in it. Good Friday and Easter Monday are derived from it | Countdown & Working Days |
| Bank holiday dates and their weekend substitution rules for England & Wales, Scotland, Northern Ireland and US federal holidays, with the one-off UK holidays since 2011 | The standard published rules rather than a hard-coded list, so any year computes: UK holidays falling at a weekend substitute to the next weekday, with Christmas and Boxing Day never permitted to collide; Northern Ireland adds St Patrick’s Day and the Twelfth (GOV.UK; nibusinessinfo.co.uk); Scotland’s St Andrew’s Day is a bank holiday from 2007 (St Andrew’s Day Bank Holiday (Scotland) Act 2007); US federal dates are observed on the Friday before a Saturday and the Monday after a Sunday, with Juneteenth from 2021 (Public Law 117-17) and Martin Luther King Jr. Day from 1986 (5 U.S.C. §6103). One-off holidays created by royal proclamation are facts about the past, not rules, so those since 2011 are listed — the Royal Wedding, the Diamond Jubilee, the VE Day anniversary move, the Platinum Jubilee, the state funeral and the Coronation — and a future one cannot be known | Countdown & Working Days |
| Distance travelled under constant acceleration is |s(t*)| + |s − s(t*)| when the turning point t* = −u/a lies inside the interval, and |s| otherwise | Derived: v = 0 at t* = −u/a, where s(t*) = −u²/2a; the motion is monotonic either side of it, so the path length is the excursion out plus the excursion back. None of the five SUVAT equations gives distance directly. Checked in suvat_model.py against the thrown-ball and braking cases | SUVAT Calculator |
| The appreciation rate and the investment return at which buying and renting tie (the “flip points”) | Found by bisection on the page’s own year-by-year series, one assumption at a time with the rest held fixed. Both net costs move monotonically in the assumption varied, so the crossing is unique when it exists inside the searched range. Not a forecast of either rate — a statement of what would have to be true for the verdict to change | Rent vs Buy Calculator |
| Transformers are rated in VA; the winding current is P/(pf·V), and secondary voltage droops from its no-load value by the regulation figure in proportion to load | Standard single-phase transformer practice: apparent power sets the copper current regardless of power factor, which is why nameplates read kVA (Fitzgerald & Kingsley; Chapman, Electric Machinery Fundamentals). Regulation is defined as (Vno-load − Vfull-load)/Vfull-load; the page applies it linearly with load fraction from a figure the reader sets, which is a first approximation to a quantity that really depends on load power factor and leakage reactance | Transformer Calculator |
| Mifflin-St Jeor predicts measured BMR within 10% for about 82% of non-obese adults and 70% of obese adults, so the projection is banded at ±10% | Frankenfield, Roth-Yousey & Compher (2005), J Am Diet Assoc 105:775–789 — the systematic comparison of predictive equations in healthy adults. The band is that 10% applied to BMR through the same day-by-day model as the central line; it says nothing about the one in five (three in ten if obese) outside it | TDEE Calculator |
| Password shelf life: with guess rates doubling every T months, a fast offline attack gets under a day after T·(bits − 1 − log₂(86400·10¹²)) months | Derived from the page’s own crack-time arithmetic under an assumed exponential growth in hardware. The doubling period is a slider, not a fact: GPU hash rates have roughly doubled every two to three years, but no law guarantees it. bcrypt’s work factor is 2cost, so each +1 doubles the time (Provos & Mazières, 1999) | Password Strength & Crack Time |
| Tea extraction follows first-order kinetics (exponential approach to a maximum) | Spiro & Jago's steady-state kinetic model (1982), the standard framework for tea infusion kinetics research; confirmed across multiple peer-reviewed studies | Tea Brewing Calculator |
| Extraction speeds up with temperature by Arrhenius, with an activation energy near 45 kJ/mol — so 80 °C water extracts at about 44% of the rate of boiling | Tea infusion kinetics studies in the Spiro tradition report activation energies for caffeine and the polyphenols in roughly the 40–50 kJ/mol band; 45 is taken as a mid-band figure rather than a measurement of any particular tea. It is the single most load-bearing assumption on the page: move it and every time shown moves with it, though the shape does not | Tea Brewing Calculator |
| A cup cools by Newton's law with a time constant near 20 minutes for an open 200 ml mug | CALIBRATED, not derived: the effective heat-transfer coefficient is chosen so a standard mug matches the ~20 minute time constant an open mug is commonly measured at (roughly 90 °C to 65 °C in ten minutes in a 20 °C room). It bundles evaporation, radiation and convection into one number, which is why it sits well above a textbook natural-convection figure — and why the real first couple of minutes run a little faster than an exponential, evaporation being the larger term near boiling | Tea Brewing Calculator |
| 65 °C is the line above which a drink counts as “very hot” | IARC Monographs Volume 116 (2016), which classified drinking very hot beverages above 65 °C as probably carcinogenic to humans (Group 2A), on oesophageal risk. The 60–50 °C “drinkable” band shown alongside it is comfort rather than a published threshold | Tea Brewing Calculator |
| Whole roasted beans pour to a bulk density near 380 kg/m³, against a solid density near 680, so a poured bed is about 56% solid | Standard figures for roasted coffee; 0.56 sits in the 0.55–0.60 band for random loose packing of irregular grains, and below the 0.64 random close-packed limit. The equivalent bean sphere (7.50 mm) is DERIVED from mass and solid density rather than typed separately — checked in models/grinder_model.py | Coffee Grinder Simulator |
| Grind size follows Bond’s comminution law, so the last halving costs far more energy than the first | The standard size-reduction relation: energy goes as the difference of the inverse square roots of the sizes. It is used here for its SHAPE, which is what turns an uneven strike count into a wide size distribution. The absolute micron scale is a calibration, not a prediction, and the page says so | Coffee Grinder Simulator |
| Deepening the bed from one bean to five collapses lid strikes per bean about fivefold | Emergent from the simulation, averaged over five seeds, not asserted: 2.03 lid strikes per bean per second one bean deep against 0.41 at five, a factor of 4.95. Bean-on-bean restitution of 0.35 means four layers of bed absorb over 99% of a strike. This reproduces the advice printed in every blade grinder manual — grind small batches — from bouncing physics | Coffee Grinder Simulator |
| Pulsing lowers the peak temperature; this model still cannot show it improving evenness | Two different questions, answered separately. On evenness, compared at equal blade-on time over ten seeds: spread 2.48 steady against 2.55 pulsed with a seed-to-seed scatter of 0.27, so the difference is a quarter of the noise — reported as unresolved rather than claimed. On heat, the same energy arriving in bursts with cooling between them peaks lower, which is arithmetic on Newtonian cooling and is where the benefit shown here comes from | Coffee Grinder Simulator |
| A burr set cannot emit a particle larger than its gap, giving a spread near 1.5× against a blade grinder’s 2.5× at the same median | The defining property of a burr grinder. Each nip also sheds a small mass fraction below 100 µm, a few per cent at filter and near a tenth at espresso, matching published distributions; by count those fines dominate and by mass they are negligible, so everything here is mass-weighted — checked in models/grinder_model.py | Coffee Grinder Simulator |
| 15 g of filter grind is about 740 J, or thirty to fifty seconds of hand cranking | Bond’s law with the constant fitted to the BLADE simulation — strike counts in a bouncing-bean model with no handle in it — applied to a hand grinder at the 15–25 W one arm sustains on a short handle. Real hand grinders do 15 g of filter in twenty to forty seconds, and 18 g of espresso in one to two minutes. A constant calibrated on one mechanism predicting another is the closest this page comes to a check against reality rather than itself | Coffee Grinder Simulator |
| CTC teabag tea reaches near-maximum extraction by ~6-8 minutes | Published tea extraction studies (De Gruyter 2022; open-access black tea extraction kinetics research) — used to calibrate this tool's teabag rate constant | Tea Brewing Calculator |
| Smaller leaf particle size extracts faster (teabag vs. loose leaf) | Price & Spiro (1985), "effect of leaf size... on the rate of infusion" — decreased particle size increases extraction rate constants | Tea Brewing Calculator |
| Closed-form heat-diffusion solution for boiling-egg cook time | Williams, C.D.H., University of Exeter, "The Science of Boiling an Egg" — verified against all four of the author's own published worked examples | Egg Timer Calculator |
| Egg white and yolk coagulation temperature ranges | Williams (as above); egg white 62-80°C depending on protein fraction, yolk 65-70°C | Egg Timer Calculator |
| Green-ring (ferrous sulphide) formation above ~77°C | Williams (as above); hydrogen sulphide from the white reacting with iron in the yolk at elevated temperature | Egg Timer Calculator |
| Poaching water temperature (82-88°C, below a rolling boil) | Cross-referenced consensus across multiple culinary references (Jessica Gavin, What's Cooking America, American Egg Board) | Egg Timer Calculator |
| Semi-infinite-slab heat conduction (one-sided pan frying) | Semi-infinite solid with a step change in surface temperature, T(x,t) via the complementary error function — Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, transient conduction chapter; reuses this tool's own verified thermal constants | Egg Timer Calculator |
| Typical pan temperatures by fried-egg style (120-165°C) | Cross-referenced culinary references (Food Republic, Tasting Table, Gygi, All-Clad) | Egg Timer Calculator |
| Lumped-capacitance exponential heating model for stirred scrambled eggs | Standard result for a well-mixed thermal mass — the same model form used in this site's Tea Brewing and Beverage Cooling calculators, calibrated against published scrambled-egg timing (All-Clad, Hestan Cue) | Egg Timer Calculator |
| Typical egg masses by type (chicken sizes, duck, quail, goose, turkey) | Compiled from multiple poultry and nutrition references (Meyer Hatchery, Cackle Hatchery, USDA weight classes) | Egg Timer Calculator |
| Thermal death kinetics of Salmonella in egg yolk (D = 0.28 min at 60.0 °C, 0.087 min at 62.2 °C, giving z = 4.33 °C) | Schuman, J.D. & Sheldon, B.W. (1997), "Thermal Resistance of Salmonella spp. and Listeria monocytogenes in Liquid Egg Yolk and Egg White", Journal of Food Protection 60(6):634–638. The z-value used is re-derived here from the paper's own two D-values, so the pair is internally consistent rather than mixed from separate studies. Important: that study measured liquid yolk held at a uniform temperature in sealed capillary tubes. It makes no claim about whole shell eggs, about thermal gradients inside one, or about any cooking time being safe — applying its figures to a modelled gradient is this site's extrapolation and is presented as such on the page, not as a finding of the authors. | Egg Timer Calculator |
| Salmonella Enteritidis prevalence in shell eggs (~1 in 20,000) | FDA Salmonella Enteritidis Risk Assessment, as cited in "Prevention of Salmonella Enteritidis in Shell Eggs During Production", Federal Register (2004). Quoted as context for why raw-egg dishes are not constantly causing illness — not as a basis for relying on the odds. | Egg Timer Calculator |
| Probability of salmonellosis per egg serving (0.00002%–0.00045%) | Joint FAO/WHO Risk Assessments of Salmonella in Eggs and Broiler Chickens, risk characterization chapter — from 2 illnesses per 10 million servings at 5% flock prevalence with reduced storage, to 4.5 per million at 50% prevalence with elevated storage. Quoted on the page as the published answer to “how likely am I to fall ill”, which this tool deliberately does NOT compute: doing so would need a starting bacterial count and a dose-response model, and the same assessment notes its output is more sensitive to storage than to either initial numbers or cooking. | Egg Timer Calculator |
| Official egg cooking guidance (63 °C/145 °F for 15 s; UK Lion-mark advice differs) | FDA Food Code for raw shell eggs broken for immediate service (68 °C/155 °F for 17 s otherwise); UK Food Standards Agency, which treats British Lion-marked eggs as low enough risk to eat runny including for vulnerable groups. Two authorities differ because they regulate different egg supplies; the page cites both rather than picking one. | Egg Timer Calculator |
| Tap drill size formula (major diameter minus pitch/1÷TPI, ~75% thread engagement) | Standard machinist reference, cross-checked against published tap drill charts for both metric and UNC threads | Screw, Stud & Tap Drill Sizing |
| Bolt proof, yield and ultimate strength by property class | ISO 898-1 — all three values for class 8.8 step at d = 16 mm, which matters here because M20 is offered. Classes 10.9 and 12.9 do not step in this range. | Screw, Stud & Tap Drill Sizing |
| Coffee-to-water ratios by brew method (1:16 pour-over, 1:15 French press, 1:2 espresso, etc.) | Specialty Coffee Association Gold Cup Standard, cross-referenced across multiple brewing guides | Tea & Coffee Brewing Calculator |
| Average human visual reaction time (~250ms, <200ms fast, <150ms exceptional) | Cross-referenced across multiple published reaction-time studies and large-scale benchmarks (Human Benchmark, ~81 million participants) | Precision Timer & Stopwatch |
| World Athletics false-start threshold (100ms) | World Athletics competition rules — reaction below this is ruled a false start on the basis that it precedes possible neurological response to the start signal | Precision Timer & Stopwatch |
| Tabata protocol (20s work / 10s rest × 8 rounds) | Izumi Tabata's 1996 study on high-intensity intermittent exercise, the origin of the widely-used training format | Precision Timer & Stopwatch |
| Web Audio API lookahead scheduling for drift-free timing | Standard technique documented by the Web Audio community (the "Tale of Two Clocks" pattern) for sample-accurate browser audio timing | Precision Timer & Stopwatch |
| Claim | Source | Tool |
|---|---|---|
| Trigonometric ratios and the Pythagorean theorem | Standard right-triangle trigonometry — foundational identities with no single citable origin, cross-referenced across any standard trigonometry reference | Trigonometry & Pythagoras |
| 2D/3D area, perimeter, volume and surface-area formulas | Standard Euclidean geometry formulas, cross-referenced across any standard geometry reference | Geometry — 2D & 3D |
| Function plotting: pixel sampling, discontinuity detection, equal-scale axes | Standard numerical plotting technique — evaluate at each pixel column and break the path where the function is undefined, or where a steep step between two columns has a midpoint value that does not lie between them, which is the signature of a pole rather than a steep curve; no single citable origin | Graphing Calculator |
| Exact distributions for dice sums and for keep-highest / keep-lowest rolls; four non-transitive dice each beating the next with probability 2/3; and a second pair of six-sided dice with the same sum distribution as an ordinary pair | Sums are computed by discrete convolution of the uniform single-die distribution; keep-highest and keep-lowest come from the order statistics of n independent uniform draws, P(max ≤ k) = (k/s)n. The four dice are Efron’s dice, introduced by Bradley Efron and popularised by Martin Gardner in Scientific American; the alternative pair is the Sicherman dice (George Sicherman, again via Gardner), which are provably the only other pair of positive-integer six-sided dice with the sum distribution of 2d6. Fairness uses Pearson’s chi-squared goodness-of-fit test against the uniform distribution at the 5% level. Randomness is the Web Crypto API’s getRandomValues with rejection sampling to eliminate modulo bias. All distributions are re-derived independently in dice_model.py in exact rational arithmetic (77 checks). Ideal dice only — no physical bias, spin or throw technique is modelled | Dice Roller & Coin Toss |
| Normal distribution, standard deviation and z-scores | Standard statistical theory. Quartiles by linear interpolation between order statistics (R type 7, also numpy's default); Tukey's 1.5 × IQR outlier fences; Freedman–Diaconis histogram bin width; Blom plotting positions with Acklam's inverse-normal approximation for the Q–Q plot; adjusted Fisher–Pearson skewness and excess kurtosis; Welch–Satterthwaite degrees of freedom for the two-sample t-test; Galton board as the physical model of the binomial and the central limit theorem. All cross-checked in stats_model.py | Statistics & Probability |
| Percentage arithmetic (X% of Y, reverse %, increase/decrease) | Standard arithmetic — percent means per hundred; same identities used by the site command palette | Percentage Calculator |
| Operator precedence, associativity and the angle mode, and the exact value a result is stored as | Precedence follows ordinary mathematical convention: brackets, postfix factorial and percent, exponentiation, unary minus, multiplication and division, then addition and subtraction. Two conventions are worth naming because calculators genuinely disagree — exponentiation is right-associative (2^3^2 = 512, not 64) and unary minus binds looser than it (−2² = −4). Evaluation is a hand-written recursive-descent parser, not eval, so malformed input is refused rather than repaired. One notation outside strict syntax is accepted deliberately: an x with a number on each side of it is a multiplication sign, so 4x4 is sixteen and 10 x 8 is eighty — that is how the notation is written on a tyre, a photo or a length of timber. It is decided in the tokeniser one token after a number, so a name like x, an expression like x^2 and a hexadecimal literal like 0x16 are all untouched; a variable you have defined as x can still be multiplied explicitly as 4*x*4. Arithmetic is IEEE 754 double precision, as in every browser, which is why 0.1 + 0.2 is 0.30000000000000004 and the page says so instead of rounding it away; round() is half-to-even, so round(2.5) is 2. calc_model.py implements the same grammar independently in Python and the two are pinned together by 77 shared expressions that must evaluate identically and 19 malformed inputs both must refuse. Those cases are driven through this page’s own JavaScript by build-registry.py --qa, so the two implementations cannot drift apart unnoticed — which they did the day the 4x4 rule was added, in two separate ways. Physical constants are the SI 2019 definitions, which are exact rather than measured | Scientific Calculator |
| Claim | Source | Tool |
|---|---|---|
| Blood alcohol is modelled with the Widmark distribution equation, a Seidl height-and-mass regression for the distribution factor, first-order gastric absorption and zero-order hepatic elimination | E. M. P. Widmark, Die theoretischen Grundlagen und die praktische Verwendbarkeit der gerichtlich-medizinischen Alkoholbestimmung, 1932; P. Seidl et al., “A theoretical approach to estimate blood alcohol concentration”, Forensic Science International, 2000. Both are population models. Neither describes any individual, and this tool makes no accuracy claim and cannot assess fitness to drive | Blood Alcohol Curve |
| Alcohol is eliminated at roughly 0.010–0.025 g per 100 mL of blood per hour across healthy adults, and the rate is not changed by coffee, food eaten afterwards, cold air, exercise or sleep | The range is the one consistently reported in forensic toxicology reviews of controlled dosing studies; the ineffectiveness of the folk remedies follows directly from elimination being enzyme-limited rather than concentration-limited. The tool exposes the whole range as a slider precisely because no individual can know their own value | Blood Alcohol Curve |
| Drink-drive prosecution limits: 80 mg/100 mL in England, Wales, Northern Ireland, the United States and Canada; 50 in Scotland, Ireland and much of Europe; 20 in Sweden, Norway and Poland; 30 in Japan; zero in Czechia, Hungary, Slovakia and Romania | Statutory limits published by each jurisdiction, read September 2026. Listed as information about the law, never as guidance that driving at any level is acceptable. They change, and nearly all of these places set lower or zero limits for learner, novice and professional drivers — verify locally | Blood Alcohol Curve |
| BMI formula and WHO adult category thresholds | World Health Organization / CDC standard adult BMI classification (Underweight <18.5, Normal 18.5–24.9, Overweight 25–29.9, Obese Class I–III) | BMI Calculator |
| Ethnicity-adjusted BMI thresholds for South/East Asian populations | National health body guidance (Japan, Singapore, India) applying lower overweight/obese cutoffs, reflecting measurably different risk curves at the same BMI | BMI Calculator |
| Waist-to-height ratio read alongside BMI, and where the two disagree | The 0.4 / 0.5 / 0.6 waist-to-height bands of UK NICE obesity guidance (2022 update), the same thresholds as this site’s waist-to-height tool. The disagreement cases are worked directly: 70 kg at 175 cm with a 92 cm waist is BMI 22.9 with a ratio of 0.53; 95 kg at 185 cm with an 86 cm waist is BMI 27.8 with a ratio of 0.46. The verdict that BMI suits populations rather than individuals is Keys et al. (1972), who named the index | BMI Calculator |
| Riegel endurance formula (T₂ = T₁·(D₂/D₁)^1.06) for running race-time prediction | Riegel, P. (1981), “Athletic records and human endurance”, American Scientist — fitted to world-record performances. Vickers, A.J. & Vertosick, E.A. (2016), “An empirical study of race times in recreational endurance runners”, BMC Sports Science, Medicine and Rehabilitation — 2,303 recreational runners: well calibrated up to the half marathon, marathon predictions at least 10 minutes too fast for half of them | Pace Calculator |
| Elite, competitive, average and beginner running times by distance (5K, 10K, half, marathon) | Cross-referenced across RunRepeat's State of US Marathons 2025, Marathon Handbook's age/ability benchmark guides, and World Athletics elite/world-record data | Pace Calculator |
| Cycling speed bands by rider category (beginner through professional) | Cross-referenced cycling references (BikeTips, Fitness Health, road-bike.co.uk) — recreational 10-15mph through professional 25-31mph sustained | Pace Calculator |
| Swimming pace bands per 100m by ability level | Cross-referenced swimming references (220 Triathlon, Polar, Fitness Health) and current 100m freestyle world records | Pace Calculator |
| Log-normal shape for race-finish-time population distributions | Standard, well-documented statistical model for right-skewed finish-time data; calibrated here against published median/beginner benchmarks rather than a specific event's raw results | Pace Calculator |
| Mifflin-St Jeor BMR equation and standard TDEE activity multipliers (1.2–1.9) | Mifflin MD, St Jeor ST, et al. (1990), American Journal of Clinical Nutrition — cross-referenced against the Frankenfield comparison review and ISSA/NASM activity multiplier tables | TDEE Calculator |
| Katch-McArdle BMR = 370 + 21.6 × lean body mass (kg) | McArdle, Katch & Katch, Exercise Physiology — the lean-mass equation, offered when a body-fat figure is given. It is only as good as that figure, and the page holds body-fat percentage constant as weight changes in the projection | TDEE Calculator |
| Exercise costed in METs, net of resting: kcal = MET − 1, times body mass in kg, times hours | Ainsworth et al. (2011), 2011 Compendium of Physical Activities: a second update of codes and MET values, Med Sci Sports Exerc 43:1575–1581 — running at 10 min/mile 9.8, cycling at 12–14 mph 8.0, swimming laps 5.8, jogging 7.0, hiking 6.0, brisk walking 5.0, easy walking 3.5, hard weights 6.0, general weights 3.5, yoga 2.5. Digestion is taken as roughly a tenth of daily energy, a round figure within the usual 8–15% range | TDEE Calculator |
| The naive 3,500 kcal/lb (7,700 kcal/kg) fixed-deficit rule overestimates real-world weight loss, increasingly over longer timeframes | Thomas DM, Martin CK, Lettieri S, et al. (2014), International Journal of Obesity; Hall KD, Sacks G, Chandramohan D, et al. (2011), The Lancet — cross-referenced against a direct simulation comparing the fixed-deficit rule to a weight-dependent BMR recalculation | TDEE Calculator |
| Claim | Source | Tool |
|---|---|---|
| Chords are stored as semitone intervals from the root; an inversion rotates that list, moves the new bass to the bottom and stacks the rest above it | Standard definitions of chord construction and inversion. Verified over all 22 qualities in all 12 keys in every inversion — 264 voicings — that the pitch classes are unchanged and the result is genuinely ascending, which rotation alone is not for chords containing a ninth | Piano Chords |
| A chord is named from a set of notes by trying each note as the root and matching the remaining intervals, with the bass breaking ties and producing slash notation | Standard chord-symbol convention. The page’s own function is driven in a browser engine against an independent Python implementation over 344 cases, and every name it produces is checked to describe exactly the notes pressed | Piano Chords |
| Some sets of notes have more than one correct name — C6 is also Am7/C, Csus2 is also Gsus4/C, and augmented triads and diminished sevenths are symmetrical so every rotation names another chord of the same type | A property of the intervals, not a limitation of the tool: verified by construction, and the alternatives are listed rather than silently discarded | Piano Chords |
| Black keys are placed by the equal-front construction — the visible front of every white key is the same width — which puts C♯ nearer C, D♯ nearer E, G♯ on the G/A boundary and A♯ nearer B | Standard keyboard geometry, with a black key 0.58 of a white key’s width, close to but not identical to any particular maker. Verified that all three fronts in the C–E group and all four in the F–B group come out equal, and that G♯ lands within 0.02 of the boundary | Piano Chords |
| Voice leading is found by searching every inversion in every reachable octave for the voicing closest to the previous chord, and never produces more movement than root position | Minimum total semitone movement, one reasonable definition of smooth among several — parallel fifths and melodic shape are not modelled, which the page states. Verified across six progressions in all twelve keys, 72 combinations, that the voiced version never moves further than root position | Piano Chords |
| Fingerings are conventional defaults for a chord played on its own: thumb on the lowest note in the right hand, little finger on the lowest in the left | Standard piano practice. Presented as defaults rather than answers, because the right fingering depends on what follows and the page does not know that | Piano Chords |
| Stave positions are computed from a diatonic index — letters counted rather than semitones — and the pitch each clef assigns to its bottom line: E4 treble, G2 bass, F3 alto, D3 tenor | Standard definitions of the G, F and C clefs. Verified against the facts each clef is named for: the G clef’s spiral is on G4, the F clef’s dots straddle F3, and the C clef points at middle C — the middle line in alto, the fourth line in tenor | Reading Music |
| Key signature accidentals are placed by rule rather than from a table: up a fifth from the previous one where that stays under a per-clef ceiling, otherwise down a fourth, and the mirror of that for flats | Traditional engraving practice. Verified against published key signatures for all four clefs and both directions — 56 accidental positions, including the treble clef’s G♯ above the top line and the bass clef’s F♭ on the fourth line, both of which look like anomalies until the rule is stated | Reading Music |
| The mnemonics offered for each clef spell that clef’s actual lines and spaces | Checked rather than assumed: every mnemonic’s letters are compared against the notes computed for that clef’s five lines and four spaces, and the words’ initials against the letters they claim to give | Reading Music |
| A key signature’s name follows from it: the last sharp plus a semitone, or the second-to-last flat, with one flat as the exception | Conventional teaching rules, verified to produce the correct key for all fourteen signatures. Middle C is C4 in scientific pitch notation and 261.63 Hz at A440 | Reading Music |
| Note values halve at each step and a dot adds half the note’s own value | Definition of the note-value system. British names (semibreve, minim, crotchet) with the American names alongside; verified that each value is exactly half the one above and divides a bar of four exactly | Reading Music |
| Metronome beats are scheduled against the Web Audio API’s own clock ahead of time rather than fired by a page timer, so scheduler lateness never reaches the output and timing error cannot accumulate | The standard Web Audio lookahead-scheduler technique, as set out in Chris Wilson’s “A Tale of Two Clocks” (Web Audio API guidance) and used by browser-based digital audio workstations. Each beat time is derived from the previous by exact addition of 60 ÷ BPM seconds; the page measures and displays its own scheduler wake-up jitter alongside the beat placement error, which is zero by construction | Metronome |
| Tempo terms (Largo, Andante, Allegro, Presto and the rest) are shown as conventional modern BPM ranges, not definitions | Conventional modern usage. The ranges overlap between sources and have shifted since the terms were adopted, which the page states rather than implying a fixed number; offered as orientation only | Metronome |
| Note-value lengths and delay times follow from 60000 ÷ BPM milliseconds per beat, with dotted values ×1.5 and triplets ×2/3 | Arithmetic definition of note values; the same figures a delay unit or a plug-in is set from | Metronome |
| Scales and modes are stored as semitone intervals from the root, and every fretboard position, chord and name is derived from those plus the open-string pitches | Standard interval definitions in twelve-tone equal temperament. The seven modes are rotations of one interval list, verified by the page reproducing C major, D dorian, E phrygian, F lydian, G mixolydian, A aeolian and B locrian from it | Scales, Modes & Fretboard |
| Seven-note scales are spelled by assigning consecutive letters from the root and computing each accidental, so each letter is used exactly once and double accidentals appear where the notation requires them | The spelling convention of Western common-practice notation. Verified against known cases: F♯ major yields E♯, E♭ major yields A♭ and B♭ with no sharps, and a key needing a triple accidental is reported as unspellable rather than approximated | Scales, Modes & Fretboard |
| Diatonic chords are built by stacking thirds within the scale itself, not looked up per key | Definition of tertian harmony. Verified against the standard results: C major gives I ii iii IV V vi vii° and C harmonic minor gives i ii° III+ iv V VI vii°, with the corresponding seventh chords | Scales, Modes & Fretboard |
| Fretboard position boxes are found by searching for hand-width fret windows holding at least two scale notes on every string, anchored on a scale note on the lowest string | Derived, not tabulated. Verified against the familiar shapes: A minor pentatonic returns five boxes at frets 5, 8, 10, 12 and 15, and a seven-note scale returns one position per degree in every key | Scales, Modes & Fretboard |
| The strumming pattern and chord sequence are scheduled against the audio hardware’s own clock ahead of time, so the spacing between strums does not wander within a bar | The same lookahead-scheduler technique the metronome uses, and for the same reason: a page timer promises “at least this long” and never “exactly this long”. The sweep across the strings and the gap between strums are separate controls because they are separate quantities — how long the pick takes to cross, and how often it crosses | Guitar Chords |
| A muted strum is the same stroke played with a much shorter decay, not a different sound | On the instrument it is the fretting hand relaxing so the strings still speak briefly; here the same Karplus–Strong pluck is given a short hold time and a lower gain | Guitar Chords |
| Chord voicings are found by exhaustive search over the fretboard rather than looked up in a chord dictionary, with the barre modelled as a finger that presses every string it spans | Derived from the tuning and the chord’s interval structure directly; no chord table is used anywhere in the tool. Verified by requiring the search to rediscover fifteen standard open and barre shapes — x32010, 320003, xx0232, 022100, 133211 and others — from pitch classes alone, each ranked first | Guitar Chords |
| Standard tunings used: guitar EADGBE, drop D, DADGAD, open G, half step down; ukulele GCEA re-entrant; bass EADG; mandolin GDAE | Conventional instrument practice. Each is stored only as its open-string pitches, and every shape shown is recomputed from those numbers — which is why a capo, which raises all of them equally, needs no separate handling | Guitar Chords |
| A note already being tracked is held to a far lower threshold than one being acquired, so a decaying string keeps reading long after it would previously have been dropped | Two thresholds with hysteresis, plus an adaptive floor that only learns from frames where nothing was detected — a floor fed by the note itself rises to meet it. Measured by bisection on the shipped detector: about −64 dB relative to a firm pluck to acquire and −84 dB to hold, a gap of 20 dB, which on a two-second decay constant is roughly five more seconds of usable reading. Checked in a quiet room and in one 30 dB noisier | Guitar Tuner |
| 0 dB SPL is 20 µPa; level from pressure is 20 log₁₀(p/p₀) and from intensity 10 log₁₀(I/I₀) | The SI definition of sound pressure level. The two forms agree because intensity goes as pressure squared — checked in models/sound_model.py, along with the reference giving exactly 0 dB | Sound Levels & Decibels |
| Point sources fall 6 dB per doubling of distance, line sources 3 dB; uncorrelated sources add as 10 log₁₀Σ10^(L/10) | Geometric spreading over a sphere (1/r²) against a cylinder (1/r), and incoherent energy addition. Free field only — the page states that reflections, barriers, ground effect and air absorption are not modelled | Sound Levels & Decibels |
| 85 dB(A) for eight hours, halving for every 3 dB above it | The ISO 1999 / EU Physical Agents (Noise) exchange rate. The US OSHA rule uses a 5 dB exchange and is markedly more permissive at high levels, which the page says | Sound Levels & Decibels |
| The microphone meter is relative (dBFS), not absolute SPL | A browser exposes a normalised sample stream from a microphone of unstated sensitivity, usually behind automatic gain control that cannot reliably be disabled — there is no pascal value to compare against 20 µPa. Differences are valid; the absolute number is not, and the page refuses to present one | Sound Levels & Decibels |
| The steadiness test judges the scatter of recent frames around a fitted straight line, not their total spread, so a peg being turned reads normally while jitter still refuses | A turning peg moves the pitch steadily in one direction and produces a large spread with a small residual; a second string ringing produces a large residual. The previous total-spread test rejected readings during the one action the tool exists to support | Guitar Tuner |
| Twelve-tone equal temperament: every semitone is a frequency ratio of 21/12, referenced to A above middle C | ISO 16:1975, Acoustics — Standard tuning frequency, which fixes A4 at 440 Hz. The tool leaves the reference editable from 415 to 466 Hz because orchestral practice commonly sits at 442–443 and historically informed performance at 415 | Guitar Tuner |
| The pitch is measured on the isolated fundamental, because a real string is stiff and its partials are not exact harmonics | String stiffness makes partial n sit at n·f0·√(1+Bn²) rather than n·f0 — the standard inharmonicity model, B being the stiffness coefficient (Fletcher & Rossing, The Physics of Musical Instruments, on stiff strings). A waveform of sharp partials does not repeat exactly at 1/f0, so a correlation over the raw signal peaks early and reads sharp: measured at 3.2 cents on a wound low E. A coarse pass locates the pitch, the signal is zero-phase band-passed around that estimate to leave the fundamental, and the peak is measured there — which brings every string inside 0.13 cents. The two stages must agree to within a semitone or the tool declines, because a large disagreement means a second string is ringing inside the same band rather than that a correction is needed. Verified in tuner_check.js and tuner_dynamics.js. | Guitar Tuner |
| A tracking window: the reading only accepts pitches a peg could have reached | Standard track gating, as used in radar and any other tracker that must not be captured by a second target. The tracked pitch may move at up to 600 cents per second — roughly four times a brisk peg turn — with the window clamped to 35–200 cents so it neither closes on detector jitter nor opens wide enough during a gap to swallow a different string. A pitch outside the window is adopted only if five consecutive frames are clean (clarity ≥ 0.85) and agree within 25 cents, which distinguishes a deliberate string change from an interruption: measured, a genuine new note reads 0.98 and is steady, while two notes together produce a composite at 0.55–0.64 that wanders tens of cents. The independence this assumes is that interference is not itself a clean sustained note near the tracked pitch — a second string tuned within a semitone of the first will be followed, and no gate of this kind can help there. Verified in tuner_dynamics.js. | Guitar Tuner |
| The displayed reading is smoothed by a one-euro filter, which cannot overshoot | G. Casiez, N. Roussel & D. Vogel, “1€ Filter: A Simple Speed-based Low-pass Filter for Noisy Input in Interactive Systems”, CHI 2012. A first-order low-pass whose cutoff rises with the measured speed of the signal: first order means the output is a weighted average of past inputs and therefore can never leave the range it has seen, and the speed term means it barely lags a change. It replaced a seven-frame median feeding an error-proportional average, which on a measured 40-cent peg turn overshot by 5.8 cents and settled 5.0 cents wrong. Parameters (minimum cutoff 0.4 Hz, β 0.18, speed low-pass 0.8 Hz) were chosen against the plucked-note harness rather than by eye. The measured speed ignores anything under 70 cents a second: on a real microphone the frame-to-frame noise alone reads as about 85 cents a second, which holds the cutoff near 15 Hz and defeats the filter, so without that dead zone a lower minimum cutoff changed the displayed spread by under 3%. | Guitar Tuner |
| Pitch is found from the period of the waveform using the McLeod pitch method — a normalised square difference function, key-maximum picking and parabolic interpolation | P. McLeod and G. Wyvill, “A Smarter Way to Find Pitch”, Proceedings of the International Computer Music Conference, 2005. Implemented directly rather than adapted; verified here against synthesised tones of known frequency, including weak and absent fundamentals, mains hum and stiff-string inharmonicity | Guitar Tuner |
| Standard guitar tuning is E2 A2 D3 G3 B3 E4, and the listed alternates (drop D, DADGAD, open G, half step down, bass, soprano ukulele) | Conventional instrument practice; every frequency shown is computed from the reference pitch by equal temperament rather than read from a table, so it follows whatever reference you set | Guitar Tuner |