Question

A ball falls vertically onto a floor with momentum $p$ and then bounces repeatedly, the coefficient of restitution is $e$. The total momentum imparted by the ball to the floor is

• A. $\rho (1+e)$
• B. $\frac{1}{1-e}$
• C. $\rho \left(\frac{1+e}{1-e}\right)$
• D. $\rho (1-\frac{1}{e})$

Answer 1

Answer: (C)

$\rho \left(\frac{1+e}{1-e}\right)$

When a particle undergoes normal collision with a floor or a wall with a coefficient of restitution $e$, the speed after collision is $e$ times the speed before the collision. Therefore, change in momentum after 1st impact is $ep-(-p)=p(1+e)$. Similarly, after the 2nd impact, the change in momentum will be $e\left(ep\right)-(-ep)=ep(1+e)$ and so on.

Therefore, total change in momentum of the ball $=$ momentum imparted to the floor $=p(1+e)(1+e+e^2+.......)=p\left(\frac{1+e}{1-e}\right)$