How does a bouncing ball work?

How does a bouncing ball work?

A bouncing ball alternates between two simple things: free fall under gravity, and a brief collision with the floor that reverses its velocity but keeps only a fraction of its speed. That fraction is the coefficient of restitution $e$: a ball with $e = 0.8$ leaves the floor at 80% of the speed it arrived with, so each bounce reaches only $e^2 = 64\%$ of the previous height.

What happens between bounces?

Once the ball leaves the floor, gravity is the only thing acting on it. The vertical velocity changes at a steady $-g$, and the horizontal velocity stays constant (there is no friction with the air in the basic model). Dropped from height $h_0$ it reaches the floor after $t_0 = \sqrt{2h_0/g}$ with speed $v_0 = \sqrt{2gh_0}$ — energy conservation gives the same result.

What does the coefficient of restitution mean?

No real collision is perfectly elastic. The ball squashes against the floor, stores some energy elastically, and returns most — not all — of it. Restitution measures how much: the ratio of separation speed to approach speed.

How high does each bounce go?

The ball hits with speed $v_0$ and leaves with $ev_0$. Rising against gravity, a launch speed $v$ reaches height $v^2/2g$, so the next bounce reaches $h_1 = (ev_0)^2/2g = e^2 h_0$. Repeating gives a geometric sequence:

Drop a ball with $e = 0.8$ from $2\ \text{m}$ and the peaks are $1.28$, $0.82$, $0.52$, $0.34$ m and so on. The kinetic energy retained per bounce is $e^2$ — for $e = 0.8$ only 64%, so after ten bounces just about 1% of the original energy is left.

Where does the lost energy go?

During impact the ball compresses like a spring and then expands, but the compression and rebound do not follow exactly the same force curve — that is called hysteresis. The area between the two curves is energy turned into heat and sound in the ball and floor. Softer, more internally lossy materials (clay, dead tennis balls) lose more; hard, springy ones (a superball) lose less.

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