Imagine two lumps of clay. You throw one at the other, and they smack together, stick, and move off as one blob. That's the classic inelastic collision. Something is clearly lost — the sound of the impact, a little heat from the deformation, and the fact that the clay is now squished. That "something lost" is kinetic energy. But here's the key: the total momentum of the two lumps before the crash is exactly the same as the momentum of the single combined lump after the crash. Momentum is always conserved in any collision if no external force acts. Energy, however, can change form.
Note
In an inelastic collision, momentum is conserved but kinetic energy is not conserved. Some kinetic energy transforms into other forms — heat, sound, or permanent deformation.
The most extreme version is a perfectly inelastic collision, where the objects stick together after impact. That's the clay example. But not all inelastic collisions are that dramatic. A car crash is inelastic — the cars crumple, metal bends, heat radiates — but they usually don't fuse into one piece. The defining feature is simply that kinetic energy is not the same before and after.
The Precise Statement
For any two objects colliding inelastically (with no external forces):
Momentum conservation:
m1u1+m2u2=m1v1+m2v2
Kinetic energy is NOT conserved:
21m1u12+21m2u22>21m1v12+21m2v22
The "lost" kinetic energy appears as heat, sound, or deformation energy. You cannot write an equation that sets the initial KE equal to the final KE — that would be an elastic collision.
For a perfectly inelastic collision (objects stick together, final velocity v is common):
m1u1+m2u2=(m1+m2)v
Why Does This Happen?
When two objects collide, forces between them do work. In an elastic collision, that work is stored and released like a spring — no permanent change. In an inelastic collision, the material deforms permanently. The work done to squash the clay or crumple the metal is not recovered as motion; it dissipates as heat. That's why the final kinetic energy is less.
A Common Exam Trap
Students often try to "conserve energy" in an inelastic collision by writing:
Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum, losing some kinetic energy as heat/sound/deformation. …
Step 1.Elastic collision: total kinetic energy before the collision equals total kinetic energy after it, in addition to momentum always being conserved. The forces involved during impact are conservative, so mechanical energy is not dissipated -- the bodies typically separate cleanly.
Step 2.Inelastic collision: total kinetic energy before does not equal total kinetic energy after; the difference ΔQ is converted into heat, sound, light, or permanent deformation. Momentum is still always conserved, since that follows from Newton's third law regardless of collision type, but the forces involved are non-conservative.
Step 3. The extreme case of an inelastic collision is the perfectly (completely) inelastic collision, where the two bodies stick together and move off with one common velocity -- this represents the maximum possible kinetic-energy loss for the given initial momenta. …