Question

A body falling freely from a given height $H$, hits an inclined plane in its path at a height $h$. As a result of this impact the direction of velocity of the body becomes horizontal. For what value of $\frac{h}{H}$ will the body take maximum time to reach the ground?

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $1$
• D. $\frac{3}{4}$

Answer 1

The correct option is A $\frac{1}{2}$

Let say from the point $O$ to $B$ time taken is $t_1$
$H-h=\frac{1}{2}g{t_1}^2\Rightarrow t_1=\sqrt{\frac{2\left(H-h\right)}{g}}$
from the point $B$ to $C$ time taken is $t_2$.
$h=\frac{1}{2}g{t_2}^2\Rightarrow t_2=\sqrt{\frac{2h}{g}}$
Total time $T=t_1+t_2=$$\sqrt{\frac{2}{g}}\left({(H-h)}^{1/2}+h^{\frac{1}{2}}\right]$

For $T$ to be maximum

$\frac{dT}{dh}=0$
$\frac{1}{2}{\left(H-h\right)}^{\left(-\frac{1}{2}\right)}\left(-1\right)+\frac{1}{2}{h}^{\left(-\frac{1}{2}\right)}=0H-h=h\Rightarrow \frac{h}{H}=\frac{1}{2}$

For detailed solution watch the next video.