MOTION & FORCES

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.

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

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

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

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

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.

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.