The coefficient of restitution measures how elastic a collision is by comparing the relative speed of separation after impact with the relative speed of approach before impact. It is represented by and normally has a value from to .
For two objects colliding along the line of impact:
If object A approaches object B, with velocities before impact and , and velocities after impact and , then:
Use velocities resolved along the line of impact and apply a consistent positive direction. The ratio uses speeds, so the final value should not be negative.
Worked example: object A approaches stationary object B at . After impact, A continues at and B moves at in the same direction. Therefore, the approach speed is , the separation speed is , and . This describes elasticity but does not establish whether momentum is conserved.
When both final velocities are unknown, combine the restitution relationship with conservation of linear momentum, provided external impulse during the collision is negligible.
| Value of | Interpretation |
|---|---|
| Perfectly elastic collision; relative speed is fully restored | |
| Partially elastic collision; some mechanical energy is dissipated | |
| Perfectly inelastic collision; no relative separation speed |
For a ball bouncing vertically from a stationary surface:
If it is dropped from height and rebounds to , then:
For and :
A common misconception is that is the percentage of kinetic energy retained. It is a speed ratio; for a ball colliding with a fixed surface, the kinetic-energy ratio is .
In an IB SEHS HL response on Newton's laws of motion, define the quantity, state the equation, identify velocities before and after impact, and show substitution with units. The coefficient itself is dimensionless.
