AP® Physics C: Mechanics · Unit 4 · Topic 4.4
Elastic Collisions
An elastic collision conserves both momentum and kinetic energy, and those two equations together force a specific pair of final velocities, no others are possible.
In any collision, momentum is conserved as long as no outside horizontal force acts during the hit. An elastic collision adds a second condition: kinetic energy is also conserved, nothing is lost to heat, sound, or deformation. One equation alone cannot pin down two unknown final velocities, but two equations can. Solving momentum conservation and energy conservation together for a stationary target gives closed-form results: v1' = ((m1-m2)/(m1+m2))v1 and v2' = (2m1/(m1+m2))v1. You do not need to derive these from scratch every time, but you should know where they come from.
A cleaner way to see the physics, instead of grinding through the algebra, is the relative velocity rule: in any elastic collision, the relative speed of approach equals the relative speed of separation. That is, v1 - v2 (before) = -(v1' - v2') (after). This falls directly out of combining the momentum and energy equations, and it is often the fastest route to a solution, especially when paired with momentum conservation instead of the messier energy equation.
This topic sits right after general momentum conservation and right before perfectly inelastic collisions, where objects stick together and kinetic energy is not conserved. Elastic collisions are the other extreme: maximum bounce, zero energy loss. Comparing the two cases, and checking limits like equal masses or a huge mass ratio, is how this topic gets tested conceptually rather than just algebraically.
Key terms
- Elastic collision
- A collision in which both momentum and total kinetic energy are conserved.
- Conservation of momentum
- Total momentum before a collision equals total momentum after, provided no net external force acts.
- Relative velocity of approach/separation
- The speed at which two objects close in before impact and move apart after, equal in an elastic collision.
- Closed-form solution
- An explicit algebraic formula for the final velocities, derived once so it can be reused.
Watch it explained
- How it worksHow do two conservation laws together pin down both final velocities in an elastic hit?
- Worked exampleFind both final velocities when a 2 kg ball at 6 m/s elastically strikes a 4 kg ball at rest.
- How it worksHow does the relative speed of approach come out equal to the relative speed of separation?
- Going furtherWhat happens when a very light ball bounces elastically off a very heavy one?
Check yourself
End of topic.