Perfectly Inelastic Collisions

Picture a moving cart slamming into a stationary one and locking together. Before, only one cart carried momentum. After, both masses move together at one shared velocity. Momentum conservation says m1·v1 = (m1+m2)·vf, so the final speed is always slower than the original speed, because now more mass is being pushed along by the same total momentum. That slowdown is the whole story in one sentence: same momentum, more mass, less speed.

Kinetic energy tells a different story. KE depends on v squared, and momentum depends on v. Spreading the same momentum over a larger combined mass always lowers the total KE. The missing energy does not vanish — it goes into crumpling metal, warming the collision point, snapping bonds, making noise. None of that shows up as motion, so it disappears from the KE tally even though total energy of the universe is still conserved. This is why momentum conservation holds in every collision, elastic or not, while KE conservation only holds in the special elastic case.

This topic sits between elastic collisions, where KE is conserved and objects bounce apart, and the general collision framework where momentum bar charts track before-and-after states. Perfectly inelastic collisions are the extreme case of inelasticity: maximum possible energy loss for a given momentum, because the objects end with zero relative velocity. Partially inelastic collisions fall somewhere between these two extremes.