Bouncing Ball Calculator
Calculate bounce height after each impact.
Enter drop height, ball type, and gravity.
Shows all bounce heights, total distance, and an exponential decay chart.
How a bouncing ball works
When a ball hits the ground, it compresses and then springs back. The ratio of the rebound speed to the impact speed is called the Coefficient of Restitution (COR) — also written as e.
COR = v_rebound / v_impact
A COR of 1.0 = perfectly elastic (no energy lost). A COR of 0 = no bounce at all (clay, putty). Real balls fall somewhere between.
Bounce height formula: h_n = h₀ × e^(2n)
Where:
- h₀ = initial drop height
- e = Coefficient of Restitution (COR)
- n = bounce number
- h_n = peak height after the nth bounce
Each bounce height is e² times the previous one. A COR of 0.83 means each bounce reaches 0.83² = 69% of the previous height.
Total distance traveled (all bounces): d_total = h₀ × (1 + e²) / (1 − e²)
Total time until the ball effectively stops: t_total = sqrt(2h₀/g) × (1 + e) / (1 − e)
Worked example — Tennis ball (COR 0.75) dropped from 2 m on Earth:
- h₁ = 2 × 0.75² = 1.125 m
- h₂ = 2 × 0.75⁴ = 0.633 m
- h₃ = 2 × 0.75⁶ = 0.356 m
- Total distance = 2 × (1 + 0.5625) / (1 − 0.5625) = 7.14 m
- Total time = sqrt(2 × 2 / 9.81) × (1.75 / 0.25) = 3.92 s
Typical COR values by ball type:
| Ball | COR | Energy retained per bounce |
|---|---|---|
| Rubber superball | 0.90 | 81% |
| Ping pong ball | 0.87 | 76% |
| Basketball | 0.83 | 69% |
| Golf ball | 0.83 | 69% |
| Volleyball | 0.80 | 64% |
| Tennis ball | 0.75 | 56% |
| Soccer ball | 0.70 | 49% |
| Squash ball | 0.45 | 20% |
| Dead ball / putty | 0.10 | 1% |
Why energy is lost: Each bounce converts some kinetic energy into heat (material deformation) and sound (the thud). The fraction of energy retained per bounce is e². COR also depends on the surface being bounced on — a concrete floor gives higher COR than a thick carpet.
Effect of gravity: The moon has 1/6 Earth gravity. A ball dropped from 2 m on the moon bounces to the same relative heights (h_n/h₀ is unchanged), but falls and rises more slowly — the total time is much longer.