Question

A body dropped freely from a height $h$ onto a horizontal plane, bounces up and down and finally comes to rest. The coefficient of restitution is $e$, the ratio of times of two consecutive rebounds :

• A. $1:e$
• B. $e:1$
• C. $1:e^2$
• D. $e^2:1$

Answer 1

Answer: (A)

$1:e$

Velocity of ball just before the collision is $v=\sqrt{2gh}$

Velocity of ball just after the collision $v_1=ev=e\sqrt{2gh}$

Time period of flight before next collision $S=0=v_1{t}_1-\frac{g{t}_1^2}{2}$

$t_1=\frac{2v_1}{g}=\frac{2e\sqrt{2gh}}{g}$

Velocity of ball after second collision $v_2=e{v}_1=e^2\sqrt{2gh}$

Time period of flight before third collision $S=0=v_2{t}_2-\frac{g{t}_2^2}{2}$

$t_2=\frac{2v_2}{g}=\frac{2e^2\sqrt{2gh}}{g}$

So $\frac{t_1}{t_2}=\frac{1}{e}$