Bouncing Ball Calculator

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), usually 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)

Both of those are infinite sums with finite answers, which is the genuinely surprising part. A ball bounces infinitely many times and stops after a measurable number of seconds, because the times shrink geometrically.

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) = 4.47 s

That total time is the closed-form limit of infinitely many bounces. The calculator adds up the real bounces and stops once they drop below about 0.5% of the drop height, so it shows a hair less (around 4.25 s here).

Typical COR values by ball type:

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 is not a property of the ball alone. It belongs to the pair, ball and surface. The same tennis ball measures noticeably higher off concrete than off a thick carpet, and a squash ball that reads 0.45 cold reads considerably higher once it has been warmed up by play, which is exactly why squash players hit the ball around before a match.

Effect of gravity: The moon has about 1/6 of Earth’s gravity. A ball dropped from 2 m on the moon bounces to the same relative heights, since h_n/h₀ does not contain g at all, but it falls and rises more slowly. Total time scales as 1/√g, so the same tennis ball takes about 2.5 times as long to settle there.