Question

A boy 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 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 $=8 km / hr$ as the relative velocity of the rain has only a vertical component with respect the man.
$V_{x}=8\, km / hr$
$2^{\text {nd }}$ case $\rightarrow \vec{V}_{R M}=\vec{V}_{R}-\vec{V}_{M}=\left(8 \uparrow-V_{y} \uparrow\right)-12 \uparrow=-4 \uparrow-V_{y} \uparrow$
$\tan 30^{\circ}=\frac{4}{V_{y}}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{4}{V_{y}}$
$V_{y}=4 \sqrt{3} km / hr$
$\Rightarrow |V|=4 \sqrt{7} km / hr$
Velocity of Rain $=8\, km / hr \uparrow-4 \sqrt{3} km / hr \uparrow$
$\Rightarrow \tan \alpha=\frac{2}{\sqrt{3}}$ from vertical.