Question

A body falls from a height 'h' on a horizontal surface and rebounds. Then it falls again and again rebounds and so on. If the restitution coefficient is $\frac{1}{3}$, the total time taken by the body to come to rest is

• A. $\sqrt{\frac{2h}{g}}$
• B. $2\sqrt{\frac{2h}{g}}$
• C. $3\sqrt{\frac{2h}{g}}$
• D. $4\sqrt{\frac{2h}{g}}$

Answer 1

The correct option is B $2\sqrt{\frac{2h}{g}}$

Total time taken by the ball to stop bouncing
$T=t_0+2t_1+2t_2+2t_3+\cdots$$=\sqrt{\frac{2h}{g}}+2e\sqrt{\frac{2h}{g}}+2e^2\sqrt{\frac{2h}{g}}+\dots \dots =\sqrt{\frac{2h}{g}}+2\sqrt{\frac{2h}{g}}(e+e^2+\dots \dots )=\sqrt{\frac{2h}{g}}+2\sqrt{\frac{2h}{g}}\frac{e}{(1-e)}=\sqrt{\frac{2h}{g}}\left(\frac{1+e}{1-e}\right)=\sqrt{\frac{2h}{g}}\left(\frac{1+\frac{1}{3}}{1-\frac{1}{3}}\right)=2\sqrt{\frac{2h}{g}}$
Hence the correct choice is (b).