Two questions, one answer each
For any collision, ask two questions. First: is momentum conserved? If the system is isolated, yes, always — we proved that last guide. Second: is kinetic energy conserved? Here the answer varies, and it is exactly what classifies the collision.
An elastic collision conserves kinetic energy too — nothing is lost to heat, sound, or deformation. Two billiard balls, or gas molecules, come very close. An inelastic collision loses some kinetic energy. The extreme case, a perfectly inelastic collision, is when the objects stick together afterward, which loses the most kinetic energy allowed while still conserving momentum.
Perfectly inelastic: they stick
The perfectly inelastic case is the easiest to solve, because after the collision the two masses share a single common velocity. Momentum conservation gives it in one step.
Perfectly inelastic collision: the combined mass moves at the momentum-weighted average of the initial velocities.
How much kinetic energy vanished? Work it out and a clean result appears: the loss depends on the masses and their relative velocity. That lost energy did not disappear — it became heat, sound, and permanent deformation. The physics of a car's crumple zone is deliberately maximizing this loss to protect passengers.
Kinetic energy lost in a perfectly inelastic collision. It vanishes only if the relative velocity is zero — that is, if there was no real collision at all.
Elastic collisions in one dimension
An elastic collision gives you two equations — momentum conservation and kinetic-energy conservation — for two unknowns, the final velocities. The algebra is fiddly, but a beautiful shortcut cuts through it: for an elastic collision, the relative velocity reverses. The objects separate exactly as fast as they approached.
The elastic-collision shortcut: relative speed of approach equals relative speed of separation. Pair this with momentum conservation to solve without touching the quadratic energy equation.
Solving the pair gives the final velocities in closed form:
The one-dimensional elastic-collision formulas. Memorizing the relative-velocity shortcut instead is often faster and less error-prone.
Read off the limiting cases, which build deep intuition. Equal masses (m_1 = m_2): the objects simply exchange velocities — this is the trick behind a Newton's cradle. A heavy object hits a light one at rest: the heavy one barely slows, the light one shoots off at nearly twice the incoming speed. A light object hits a heavy one at rest: the light one bounces almost straight back, the heavy one barely stirs — like a ball off a wall.
The spectrum: coefficient of restitution
Real collisions live between the perfectly elastic and perfectly sticky extremes. The coefficient of restitution e is the single number that places a collision on that spectrum: it is the ratio of relative speed after to relative speed before.
Coefficient of restitution. e = 1 is perfectly elastic (relative velocity reverses fully); e = 0 is perfectly inelastic (they move together); real balls fall in between.
Two dimensions and a worked problem
In a two-dimensional collision — a cue ball striking another at an angle — momentum is a vector, so it is conserved component by component: the total x-momentum before equals the total x-momentum after, and likewise for y. You get one conservation equation per direction. A famous result: when a moving ball elastically strikes an identical stationary ball off-center, the two always separate at exactly 90^\circ.