Question

A tennis ball is released from height $h$ above ground level. If the ball makes inelastic collision with the ground, to what height will it rise after third collision, e is the coefficient resitiution between ball and ground?

• A. $h{e}^6$
• B. $e^{2}h$
• C. $e^{3}h$
• D. $Noneofthese$

Answer 1

The correct option is A $h{e}^6$

Given that,

Height = $h$

Final velocity = $v$

Initial velocity = $u$

Coefficient restitution = $e$

We know that,

Restitution = the square root of the ratio of the kinetic energy before and after collision

$e=\frac{v}{u}=\sqrt{\frac{K_1}{K_0}}$

$K_1=e^2{K}_0$

$K_2=e^2{K}_1$

$K_2=e^4{K}_0$

Similarly,

$K_3=e^4{K}_2$

$K_3=e^6{K}_0$

We know that,

The kinetic energy is

$mgh=K$

$h=\frac{K}{mg}$

The ball is initially dropped from the height $h$and acquires a kinetic energy $K_0$ when it hits the ground, before the first collision.

$mgh=K_0$

Now, If$h_1$, $h_2$and $h_3$ are the heights after the first, second and third collisions

$h_1=\frac{K_1}{mg}=e^2\frac{K_0}{mg}=e^{2}h$

$h_2=e^{4}h$

$h_3=e^{6}h$

Hence, the height raised by the ball after third collision is $e^{6}h$