## Bouncing Balls

When a ball is dropped from a height and hits the ground, it bounces and starts moving up. The speed with which it bounces up is not quite as big as the speed with which it hits the ground,

If the ball hits the ground with a speedand the coefficient of restitution between the ball and the ground isthen the ball bounces with a speedIf we ignore air resistance, and the ball is dropped from a heightthen the gravitational potential energy of the ball is initiallyand as the ball falls, this energy is changed into kinetic energy,so that just before it hits the ground, all the gravitational potential energy is converted into kinetic energy. We can write

After the ball hits the ground, it bounces with a speed

We can use the principle of conservation of energy to find the height h-1 reached by the ball:

After the ball hits the ground again, it will rebound with a speedand we can use the principle of conservation of energy to find the height reached by the ball after the second bounce.

In general, after the nth bounce, the ball will rebound with a speedand bounce to a height of

The total distance travelled by the ball is then(since after falling to hit the floor, the ball has to travel the same distance up then down again)

Ignoring the first termthis is a geometric series with first termand common ratioso the total distance travelled can be found using the formula for the infinite sum of a converging geometric series

Now addto get the total distance travelled by the ball from the instant it is dropped to the instant it comes to rest.