Elastic Collisions

In any collision, momentum is conserved. In an elastic collision, kinetic energy is conserved too. That second condition is what makes elastic collisions special: one equation (momentum) with two unknown final velocities can't be solved alone, but add the energy equation and you have two equations for two unknowns. Solving that system once and for all gives two clean formulas for a mass m1 moving at v hitting a stationary mass m2: the incoming ball ends up at v1' = (m1 - m2)v/(m1 + m2), and the ball that was at rest ends up at v2' = 2 m1 v/(m1 + m2). You can rederive these any time from conservation of momentum and conservation of kinetic energy, so you never have to memorize them cold.

The mass ratio decides everything about the outcome. Equal masses (m1 = m2) make v1' come out to zero: the cue ball stops dead and hands off all its momentum and energy to the target ball, exactly what you see on a pool table. If m1 is much smaller than m2, like a ball bouncing off a wall, v1' approaches -v: the light object just bounces straight back at the same speed, while the heavy object barely moves. If m1 is bigger than m2, the incoming ball keeps moving forward, just slower.

This topic sits next to inelastic collisions, where kinetic energy is lost to heat, sound, or deformation and only momentum conservation applies. A real collision that bounces apart isn't automatically elastic: bounciness just means the objects separate rather than stick, but some energy can still be lost in the process. True elastic collisions are an idealization, closely approached by things like billiard balls or gas molecules, but worth checking with the energy equation before assuming it holds.