AP® Physics C: Mechanics · Unit 4 · Topic 4.4
Perfectly Inelastic Collisions
In a perfectly inelastic collision, two objects stick together and move as one, so momentum conservation alone determines their shared final velocity. Kinetic energy is not conserved in this type of collision.
A cart moving at 4 m/s that crashes into a stationary cart and sticks to it can never come out of the wreck moving at 4 m/s. The stuck-together mass is bigger now, and no outside push arrived to add speed, so the common velocity has to land somewhere below the fastest incoming speed and above zero. Momentum conservation is what fixes the exact number: m1 v1 + m2 v2 = (m1 + m2) vf. Since the objects end up with one shared velocity, this one equation is all you need. There is no second unknown and no need to track forces or contact times.
Once you have vf, kinetic energy tells a different story. Add up (1/2)m1 v12 + (1/2)m2 v22 before the collision, then compute (1/2)(m1+m2)vf2 after. The after number is always smaller. That gap is the kinetic energy destroyed, and it does not vanish from the universe, it converts to heat, sound, and permanent deformation of the carts. Energy is still conserved overall, just not in the kinetic form you started with.
This topic is the extreme end of the collision spectrum that started with elastic collisions, where both momentum and kinetic energy survive untouched. Perfectly inelastic collisions sit at the opposite pole: momentum survives, kinetic energy takes the maximum possible hit for the given masses and speeds. Real collisions usually land somewhere between these two limits, so mastering both endpoints is what lets you judge any messier crash in between.
Key terms
- Perfectly inelastic collision
- A collision where the colliding objects stick together and move with one common final velocity.
- Conservation of momentum
- The total momentum of a closed system before a collision equals the total momentum after.
- Kinetic energy destroyed
- The difference between total kinetic energy before and after a collision, converted to heat, sound, and deformation.
- Common final velocity
- The single shared velocity of two objects that stick together after colliding, found from momentum conservation.
Watch it explained
- How it worksHow does momentum alone pin down the final velocity when two objects stick?
- Worked exampleFind the common speed after a 3 kg cart at 4 m/s hits a stationary 5 kg cart and sticks to it.
- Worked exampleFind how much kinetic energy that same sticking collision destroyed.
- Common mistakeWhy isn't energy conservation violated when kinetic energy disappears in a crash?
Check yourself
End of topic.