Question 255: Centre of Mass and Rotational Motion

Question

A body starts falling freely from height $$H$$ hits an inclined plane in its path at height $$h$$. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of $$\frac{H}{h}$$ for which the body will take the maximum time to reach the ground is __________.

Accepted Answer

Total time of flight $$=\mathrm{T}$$
$$T=\sqrt{\frac{2 h}{g}}+\sqrt{\frac{2(H-h)}{g}}$$
For max. time $$=\frac{\mathrm{dT}}{\mathrm{dh}}=0$$
$$\begin{aligned}
& \sqrt{\frac{2}{\mathrm{~g}}}\left(\frac{-1}{2 \sqrt{\mathrm{H}-\mathrm{h}}}+\frac{1}{2 \sqrt{\mathrm{h}}}ight)=0 \
& \sqrt{\mathrm{H}-\mathrm{h}}=\sqrt{\mathrm{h}} \
& \mathrm{h}=\frac{\mathrm{H}}{2} \Rightarrow \frac{\mathrm{H}}{\mathrm{h}}=2
\end{aligned}$$

Centre of Mass and Rotational Motion

Integer / Numeric Answer Type