Question

A man moving with $5m{s}^{-1}$ observes rain falling vertically at the rate of $10m{s}^{-1}$. Find the speed and direction of the rain with respect to ground.

Answer 1

${\overrightarrow{v}}_{r,m}=-10\hat{j}m{s}^{-1},{\overrightarrow{v}}_m=5\hat{i}m{s}^{-1}$
${\overrightarrow{v}}_{r,m}={\overrightarrow{v}}_r-{\overrightarrow{v}}_m$
${\overrightarrow{v}}_r={\overrightarrow{v}}_{r,m}+{\overrightarrow{v}}_m=(-10\hat{j})+\left(5\hat{i}\right)m{s}^{-1}$
$\Rightarrow |{\overrightarrow{v}}_r|=5\sqrt{5}m{s}^{-1}$
Inclination of the rain with vertical, $tan\theta =\frac{1}{2}$
$\Rightarrow \theta =ta{n}^{-1}\frac{1}{2}$