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Bouncing ball
Intermediate
Each bounce retains a factor e of speed.
h_n = e^{2n} h₀Each bounce multiplies the speed by e and the height by e². The height after n bounces is hₙ = e²ⁿh₀, and the total time is a geometric series.
h₀ = 1 m and e = 0.8: h₁ = 0.64 m.
The ball stops in finite time even though it bounces infinitely many times.
In more detail
A bouncing ball loses some of its kinetic energy in each impact, so its bounce height decreases. The coefficient of restitution gives the ratio of rebound speed to impact speed.
Real-world uses
- Tennis ball pressure is set to achieve a standard bounce height.
- Ball manufacturers test coefficient of restitution by dropping balls.
- A bouncing rubber ball shows smaller and smaller bounces, which can be used to estimate the coefficient of restitution.
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Frequently asked questions
Why does a ball lose height?
Because some kinetic energy is lost to deformation and heat at each bounce.
Does the bounce height depend on mass?
Not directly, only through the coefficient of restitution and drop height.
Why does each bounce get lower?
Each impact loses kinetic energy, so the speed after each bounce is a fixed fraction of the speed before.
Related topics in Momentum & Collisions
Collisions in two dimensions · Billiard ball collisions · Ballistic pendulum · Centre of mass · Velocity of centre of mass · Centre-of-mass frame