Question

If a ball is initially dropped from height h and has coefficient of restitution 0.9 .How many bounced will the ball make before comping to rest? Please help...

Answer 1

See below

Explanation:

If the ball was dropped from a height `h`, then it originally had gravitational potential energy relative to ground, of:

• `U = m g h`.

So, just before it's first collision with ground, the ball will have kinetic energy `T`, such that:

• `T = mgh`

The Coefficient of Restitution, `e`, as it applies between 2 bodies [here, ground and ball], is related to kinetic energy, as:

`e= sqrt ( {sum T_("post-collision"))/{sum T_("pre-collision")} )`

In the reference frame of ground, only the ball has Kinetic Energy.

`implies e= sqrt ( (T')/T ) = 0.9`

`implies T' = T * 0.9^2`

`implies mg h' = mgh * 0.9^2`

`implies h' = 0.81 h`

And so `h'` is the height to which the ball will rise after the first collision with ground

So:

• `h_1 = 0.81 h_o`

Follows that:

• `h_2 = 0.81 h_1 = 0.81^2 h_o`

• `h_3 = 0.81 h_2 = 0.81^3 h_o`

• `...`

• `h_n = 0.81^n h_o`

In this highly idealised model, the ball will never come to rest, ie `h_n = 0` requires that `n to oo`.