Every bounce loses energy
Drop a ball and it never quite returns to the height you dropped it from. Part of its energy turns into heat, sound and the deformation of the ball and the floor. The coefficient of restitution, written e, measures how much of the speed survives each impact: e = speed after ÷ speed before. Since height depends on the square of speed, each bounce reaches e² times the previous height.
The formulas
For a ball dropped from a height h₀:
- Height after bounce n: h₀ × e²ⁿ.
- Speed at the first impact: √(2 g h₀).
- Total distance until rest: h₀ × (1 + e²) ÷ (1 − e²).
- Total time until rest: √(2h₀ ÷ g) × (1 + e) ÷ (1 − e).
- Energy lost at each bounce: 1 − e² of the energy before the impact.
The number of bounces is infinite in theory, but the total distance and time are finite, because the bounces shrink in a geometric series.
Worked example
A basketball with e ≈ 0.846 dropped from 6 feet:
- After the first bounce: 6 × 0.846² = 4.29 ft.
- After the third bounce: 2.20 ft.
- Speed at the first impact: 13.4 mph (19.6 ft/s).
- Total distance traveled: about 36.2 ft.
- Total time until it comes to rest: about 7.3 seconds.
- Energy lost at each bounce: 28.4%.
- About 14 bounces before the height falls below 1% of the drop.
Typical coefficients
| Ball, on a hard floor | Coefficient of restitution |
|---|---|
| Superball | about 0.9 |
| Table tennis ball | about 0.87 |
| Basketball | about 0.82 to 0.87 |
| Tennis ball | about 0.73 to 0.76 |
| Baseball | about 0.5 to 0.55 |
| Lump of clay | close to 0 |
Sports rules often specify the bounce directly. A regulation basketball dropped from 6 feet must rebound 49 to 54 inches. A tennis ball dropped from 100 inches onto concrete must rebound 53 to 58 inches.
Measuring e yourself
Drop the ball from a measured height onto the surface you care about and film the bounce next to a tape measure. Then e = √(rebound height ÷ drop height). Repeat a few times and average. You can also time the bounces by sound: the time between bounces shrinks by a factor e each time.
What the model leaves out
The formulas assume a constant e and ignore air resistance, which is reasonable for heavy balls and short drops. In reality, e decreases slightly as impact speed increases, a light ball like a table tennis ball is slowed noticeably by air, and spin can change the direction of each bounce. Temperature matters too: a cold squash ball barely bounces, which is why players hit it around before a match.