Question

A man running on a horizontal road at $8$ $km/h$ finds the rain falling vertically. He increases his speed to $12$ $km/h$ and finds that the drops make angle ${30}^o$ with the vertical. Find the speed and direction of the rain with respect to the road.

Answer 1

This shows that the horizontal component of rain $=8$ $km/hr$ as the relative velocity of the rain has only a vertical component with respect the man.
$V_x=8$ $km/hr$
$2^{nd}$ case $\to {\overrightarrow{V}}_{RM}={\overrightarrow{V}}_R-{\overrightarrow{V}}_M=(8\uparrow -V_y\uparrow )-12\uparrow =-4\uparrow -V_y\uparrow$
$\tan {30}^o=\frac{4}{V_y}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{4}{V_y}$
$V_y=4\sqrt{3}$ $km/hr$
$\Rightarrow |V|=4\sqrt{7}$ $km/hr$
Velocity of Rain $=8$ $km/hr\uparrow -4\sqrt{3}$ $km/hr\uparrow$
$\Rightarrow \tan \alpha =\frac{2}{\sqrt{3}}$ from vertical.