Question

A boy running on a horizontal road at $8km/h$ finds the rain falling vertically. He increases his speed to $12km/h$ and finds that the drops makes $30^{\circ }$ with the vertical. The speed of rain with respect to the road is.

• A. $4\sqrt{4}km/h$
• B. $9\sqrt{7}km/h$
• C. $12\sqrt{7}km/h$
• D. $15\sqrt{7}km/h$

Answer 1

Answer: (A)

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