The coefficient of restitution (e) is a dimensionless number between 0 and 1 that characterizes the elasticity of a collision. It is defined as the ratio between the relative velocity of separation and the relative velocity of approach. While not strictly in the curriculum, this concept is a classic in physics experiments.
Discover FizziQ
How to measure it in class
With the FizziQ app, several methods allow measurement of the coefficient of restitution.
Method 1: Acoustic timer
- Drop a ball from height h1
- FizziQ detects successive bounces using the sound level meter
- Measure the time intervals between bounces
- Calculate successive heights and deduce e
Method 2: Video tracking
- Film the ball bounces
- Track positions at each bounce
- Measure h1 (starting height) and h2 (height after first bounce)
- Calculate e = square root of (h2/h1)
Scientific activities on this topic
- Measuring e for different balls: https://www.fizziq.org/en/activities/bouncing-ball/
- Comparison of tennis, ping-pong, golf, petanque balls
- Effect of temperature on e
Learn more
Definition:
For a collision between two objects: e = |v2’ - v1’| / |v1 - v2|
where v1, v2 are velocities before the collision and v1’, v2’ after.
Special cases:
| e | Type of collision | Kinetic energy |
|---|---|---|
| 1 | Perfectly elastic | Conserved |
| 0 < e < 1 | Inelastic | Partially lost |
| 0 | Perfectly inelastic | Maximum loss |
For a vertical bounce:
When a ball bounces on the floor (infinite mass): e = v_after / v_before = square root of (h2/h1)
Typical values:
| Ball | Coefficient e |
|---|---|
| Superball | 0.90 |
| Golf ball | 0.85 |
| Tennis ball | 0.75 |
| Basketball | 0.70 |
| Steel ball on steel | 0.95 |
| Modeling clay | approximately 0 |
Formula
General definition: e = |v_relative_after| / |v_relative_before|
For a bounce on the floor: e = square root of (h2/h1)
Height at n-th bounce: h_n = h1 x e to the power of (2n)
Kinetic energy lost: Delta Ec / Ec = 1 - e squared
Application examples
- Tennis balls are tested: e must be between 0.73 and 0.76 for certification
- Petanque balls have a very low e to stay in place
- Airbags absorb collision energy (e approximately equals 0)
- Trampolines have an e close to 1
FAQ
Q: How to measure e with FizziQ’s acoustic timer? A: Record the sound of bounces. The time between two bounces t_n is related to h by h = (1/2)g(t_n/2) squared. From h1 and h2, calculate e = square root of (h2/h1).
Q: Why can e not exceed 1? A: That would violate energy conservation: the ball would bounce higher than it started!
Q: Does e depend on velocity? A: Slightly. At high velocity, deformations can become plastic and e decreases. This effect is visible for very fast balls.
Q: Does a warm ball bounce better? A: Generally yes. Rubber is more elastic when warm. That is why tennis players roll the ball before serving.
Related concepts
Elastic Collision - Inelastic Collision - Kinetic Energy - Conservation of Energy - Momentum - Acoustic Timer