COLLIDE
The CM worldline runs straight through the wreck — that line IS momentum.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Two carts on a frictionless track, one gap between them, one dial that matters: the restitution e, from perfectly sticky (0) to perfectly bouncy (1). Everything about the aftermath is already decided — by three numbers.
Momentum
Before touching the collision, bank the invariant: p = m₁u₁ + m₂u₂. The crash is all internal forces — equal and opposite by Newton’s third law — so the center of mass glides at v_cm before, during, and after, entirely unimpressed.
Solve
Momentum gives one equation; the restitution definition e = separation/approach speed gives the second. Two equations, two unknowns, no calculus — the exit velocities drop out in closed form for every e from stick to bounce.
Collide
The crash in real time, and below it the spacetime receipt: worldlines kink at contact, slopes trade according to the formulas — but the dashed center-of-mass line runs perfectly straight through the wreck. That straight line IS momentum conservation.
Audit
Close the books: momentum identical before and after, to machine precision, at every e. Kinetic energy pays the toll ΔKE = ½μv_rel²(1−e²) — zero for the elastic bounce, maximal for the sticky one — and the ledger states exactly where it went.