Elastic and inelastic collisions

Momentum conservation follows from Newton's third law alone — whatever force one colliding object exerts on the other, the other exerts back equally and oppositely, and that symmetry holds regardless of what the objects are made of or what happens to their kinetic energy. Momentum is therefore conserved in every collision, full stop, elastic or not.

Kinetic energy has no such guarantee. Collisions can convert kinetic energy into heat, sound, or the work of permanently deforming the colliding objects, and that conversion only ever runs one way — energy leaves the "kinetic" column and does not come back. A collision that conserves kinetic energy exactly is called elastic, and it is the special case, not the default; billiard balls and gas molecules colliding come close, but two lumps of clay colliding and sticking together do not, because deforming and merging the clay consumes real energy.

A perfectly inelastic collision is the extreme of that spectrum: the objects stick together and move off with a shared final velocity, and it loses the maximum kinetic energy consistent with momentum still being conserved — not all of it, and momentum conservation is exactly what limits how much can be lost. Sticking together is what happens when the objects share the least kinetic energy compatible with a common final velocity; two clay lumps of equal mass colliding head-on at equal speed can even end up momentarily at rest, all their kinetic energy gone, while momentum — which was zero to begin with — is still conserved at zero. Between the elastic and perfectly inelastic extremes sits every real collision, losing some intermediate amount of kinetic energy while never losing any momentum at all.