NEWTON’S SECOND LAW

Constant acceleration, drawn three ways: v is the slope of the x–t curve, Δx is the area under v–t. The two kinematic equations and the calculus that ties them together.

Two masses, one rope, one pulley — the tension tells the truth.

A parabola, audited — range, apex, and time of flight in closed form.

Aim straight at the falling monkey — the ½gt² cancels and the dart always connects.

Range up a ramp: the launch that bisects slope and vertical, θ* = 45° + α/2, exact.

Slides or holds? Friction renders its verdict before anything moves.

Terminal velocity: the truce between weight and air, one exponential.

A circle’s shadow — the phasor shows why everything oscillates in cosines.

The period carries no m — mass cancels before the physics even starts.

v⃗·a⃗ = 0, bitwise — a force that steers forever and spends nothing.

Archimedes’ crown test: two weighings and the density has nowhere to hide.

Δv = vₑ·ln(m₀/m) — every m/s borrowed from the exhaust, and the ln collects.

ma + cv + kx = 0 at every drawn frame — ring, settle, or crawl.

v₀ = √(rg·tanθ) — the speed at which the road itself steers.

T² = a³, equal areas proven as an identity — Newton’s arithmetic, checked.

Shell theorem, drawn: g climbs straight to the surface, then falls as 1/r². Zero at the core.

v_esc = √(2GM/R) — the energy line that just clears the rim. Mass and angle need not apply.

v²r = GM, E = −GMm/2r, and 2KE + PE = 0 to 1e-12 — the virial ledger, audited.

a = g·sinθ/(1+β) — shape alone runs the race. Mass and radius cancel, bitwise.

τ = Iα, θ = ½αt², KE = τθ — F = ma in a spinning alphabet, audited to 1e-12.

Gravity turns the computed angular-momentum vector: full Euler–quaternion motion, exact regular precession, nutation, and an honestly bounded fast-top estimate.

Huygens’ tautochrone: beads from any height reach the bottom in the same instant — exact SHM in arc length.

A point on a rolling wheel traces a cycloid — contact stops dead, the top races at 2v, the flange runs backward.

Drop a stone through the Earth: it oscillates in 84 min — the exact period of a skimming satellite, and any straight tunnel keeps the same clock.

The gravity-train period depends ONLY on density, 2π√(3/4πGρ) — a whole planet or a marble of the same stuff take identical time.

Push a damped spring: displacement peaks BELOW ω₀, velocity peaks AT ω₀ — the exact curve acres draws for RLC current.

Two carts, two normal modes — ω_s=√(k/m) in step, ω_a=√((k+2k_c)/m) opposed. Start one and their motion envelopes trade prominence and return.

One speed, every angle — the shots fill a region capped by a parabola, y = v²/2g − gx²/2v². Anything above it is safe, forever.

Straight across is fastest (but you drift); ferry upstream for zero drift at more time. And if the current wins, you can’t land straight across.

A toppling chimney rotates, so its tip accelerates at (3/2)g·sinθ — past 42° that beats free fall, and the mortar tears at L/3 up.

Match the periods from two pivots and g = 4π²L/T² — the moment of inertia cancels completely. Kater weighed gravity without knowing I.