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Rules of Thumb

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Rule of the day · The Rulebook

45° for range — in a vacuum

Without air drag, 45° maximizes projectile range; with drag, the real optimum is nearer 30–40°.

Open the full Rulebook Exact tool: Projectile Motion & Golf Ball Flight
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One answer sets off the next

Fifteen chains through the real calculators - and every link opens the next tool with the numbers already in it.

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DIY · steak timer A steak, in cross-section The tool’s own heat equation on a coarser grid, turned every three minutes and pulled at 40.6 °C — then the core climbs to 56 on the plate. Fourteen minutes in eight seconds. core · pull point Open full simulator →
Physics · beverage cooling Can against bottle Same 330 mL of lager, same −18 °C freezer. The bottle is cold about two minutes sooner: its glass barely matters next to the still air outside, and it has 10% more surface. both temperatures · elapsed Open full simulator →
Music · guitar chords Four chords, strummed Real fret spacing, the tool’s own tuning and its default rhythm — D · D U · U D U at 88 bpm, the hand moving between shapes. Silent here; the tool plays it. chord · notes Open the chord finder →
Environmental · world atlas Every country, true to size Equal Earth, shaded by population rank, walking the eight biggest. The map data loads only when this one is dealt. population rank Open the atlas →
Physics · pendulum Double & triple pendulum Lagrangian dynamics, RK4-integrated, each pendulum run twice from starts a hair apart. For a few seconds they are one drawing; then chaos pulls them apart. twins apart Open full simulator →
Physics · mass - spring - damper Oscillation & coupling One mass rings down inside its own decay envelope; two coupled masses hand the energy back and forth. Two pulls each, on one time axis. damping · coupling Open full simulator →
Physics · collisions Newton's cradle Slightly lossy collisions, e ≈ 0.985. One ball in, one out — or two in, two out — with each ball’s momentum drawn underneath it. e ≈ 0.985 Open full simulator →
Control · ball and beam PID balancing Three setpoint steps, three tunings — underdamped, critically damped, overdamped — on one plot, so you can see what the gains do. three tunings Open full simulator →
Physics · projectile Golf ball flight Drag and Magnus lift integrated in real time: a 167 mph drive against the same drive with no spin. The backspin is worth about 120 yards of carry. carry · no spin Open full simulator →
Mathematics · solids Solids and their nets A polyhedron unfolding flat along its own edges and folding back — the same fold tree the full tool uses, faces, edges and vertices counted. assembled Open full calculator →
Physics · orbital mechanics Two orbits, to scale A 400 km circle and a Molniya ellipse, 600 × 39,900 km, positioned by Kepler’s equation. Twelve equal slices of time sweep twelve equal areas. Molniya · 12 h lap Open orbital mechanics →
Physics · relativity The light clock Two identical clocks, one carried past at 0.6c. Its photon has further to go at the same speed of light, so it ticks slower by exactly γ = 1.25. 0.6c · γ = 1.25 Open relativity →
Materials · periodic table An atom, filled in order One of the first eighteen elements, its electrons going on one at a time — two, then eight, then eight. A full outer shell lights up: the ones that react with almost nothing. H · Z = 1 Open the periodic table →
Materials · molecules Molecules, turning Water, methane, benzene, glucose and a buckyball in ball-and-stick, from the tool’s own geometry — bonds pushed apart until they stop moving. loading the tool’s geometry Open the molecule viewer →
DIY · room visualiser One paint, two walls A corner of a room with a window on one side. The same paint on the lit wall and the shaded one — the thing a flat swatch cannot show you. Sage · lit wall vs shaded wall Open the room visualiser →
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Dealt fresh on every visit, and running the same verified physics as the full tools — click a mini to feature it above, or open the simulator for controls and scenarios.

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Materials and Elements

The properties of the stuff itself — every element and what it is actually for, how a fluid behaves as it warms, and what really sets the strength of a concrete mix.

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Electrical

Circuits and magnetics - Ohm's law, power factor correction, transformer turns ratio, and wire sizing.

Open Electrical
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Mathematics

A scientific calculator that shows its own order of operations, trigonometry, a full 2D/3D geometry suite, and statistics with a genuinely random Galton board.

Open Mathematics
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Health

BMI on a full colour-coded chart, running/cycling/swimming pace, and calorie needs from a validated formula.

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Music and Acoustics

An instrument is a physics experiment you can hold. Notes named on the stave, pitch read to a fraction of a cent, chord shapes and scale positions worked out from the tuning itself, and beats placed on the audio clock rather than the browser's.

Open Music and Acoustics

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