Bounce Height Calculator (Coefficient of Restitution)
Height of a ball after any number of bounces, the total distance and time until it stops, and the coefficient of restitution from a drop test.
Results
The dashed line marks the value you entered.
What this tool does
Each bounce sends back a fixed fraction e of the speed, and because height goes with speed squared, the height shrinks by e² every time: h × e^(2n). An e of 0.75 gives a rebound height of 56 %, not 75 %, which is why e is best measured as the square root of the rebound height over the drop height. The infinite string of bounces adds up to a finite distance, h(1 + e²) ÷ (1 − e²), and a finite time: the ball is at rest after √(2h ÷ g) × (1 + e) ÷ (1 − e). The model assumes a constant e on a hard, flat surface; real balls lose a little more at higher impact speeds.
Formula
h_n = h × e^(2n) · s = h (1 + e²) ÷ (1 − e²) · t = √(2h ÷ g) × (1 + e) ÷ (1 − e) · e = √(h₂ ÷ h₁)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
h | Drop height | m |
k | Coefficient of restitution e (0 to 1) | — |
n | Number of bounces | — |
hm | Height of the first rebound (to measure e) | m |
HN | Height reached after those bounces | m |
DT | Total distance travelled until it stops | m |
TT | Time until it stops bouncing | s |
EM | Coefficient of restitution from the two heights | — |
Worked example
- Drop height2 m
- Coefficient of restitution e (0 to 1)0.75
- Number of bounces3
- Height reached after those bounces0.356 m
- Total distance travelled until it stops7.14 m
- Time until it stops bouncing4.47 s
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.