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 H/h for which the body will take the maximum time to reach the ground is____.
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Solution
Sol.
Total time of flight $=\mathrm{T}$
$$
T=\sqrt{\frac{2 h}{g}}+\sqrt{\frac{2(H-h)}{g}}
$$
For max. time $=\frac{\mathrm{dT}}{\mathrm{dh}}=0$
$$
\begin{aligned}
& \sqrt{\frac{2}{g}}\left(\frac{-1}{2 \sqrt{\mathrm{H}-\mathrm{h}}}+\frac{1}{2 \sqrt{\mathrm{~h}}}\right)=0 \\
& \sqrt{\mathrm{H}-\mathrm{h}}=\sqrt{\mathrm{h}} \\
& \mathrm{~h}=\frac{\mathrm{H}}{2} \Rightarrow \frac{\mathrm{H}}{\mathrm{~h}}=2
\end{aligned}
$$
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