Question

Two balls of same mass are dropped from the same height $h$, on to the floor. The first ball bounces to a height $h/4$, after the collision and the second ball to a height $h/16$. The impulse applied by the first and second ball on the floor are $I_1$ and $I_2$ respectively. Find the ratio $\frac{I_1}{I_2}$.

Answer 1

Velocities just before collision with ground is $u_1,u_2$ and just after collision is $v_1,v_2$

$u_1=\sqrt{2gh}$ ,$u_2=\sqrt{2gh}$

$v_1=-\sqrt{\frac{2gh}{4}}$ , $v_2=-\sqrt{\frac{2gh}{16}}$

$v_1=-\frac{\sqrt{2gh}}{2}$ , $v_2=-\frac{\sqrt{2gh}}{4}$

$I_1=m{u}_1-\left(m{v}_1\right)$

$m\sqrt{2gh}+\frac{m\sqrt{2gh}}{2}=\frac{3}{2}m\sqrt{2gh}$

$I_2=m{u}_1-\left(m{v}_2\right)$

$=m\sqrt{2gh}+\frac{m\sqrt{2gh}}{4}=\frac{5}{4}m\sqrt{2gh}$

$\frac{I_1}{I_2}=\frac{\frac{3}{2}}{\frac{5}{4}}=\frac{12}{10}=1.2$