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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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