Question

A man is coming down an incline of angle ${30}^o$. When he walks with speed $2\sqrt{3}m{s}^{-1}$ he has to keep his umbrella vertical to protect himself from rain. The actual speed of rain is $5m{s}^{-1}$. At what angle with vertical should he keep his umbrella when he is at rest so that he does not get drenched?

Answer 1

Velocity of rain w.r.t man
${\overrightarrow{v}}_{r;m}=x\hat{j}={\overrightarrow{v}}_r-{\overrightarrow{v}}_m$
${\overrightarrow{v}}_m=2\sqrt{3}[\cos 30\hat{i}+\sin {30}^o\hat{j}]$
$=3\hat{i}+\sqrt{3}\hat{j}$
${\overrightarrow{v}}_r=-3\hat{i}+(x-\sqrt{3})\hat{j}$
$5=\sqrt{3^2+(x-\sqrt{3}{)}^2}$
$16=(x-\sqrt{3}{)}^2\Rightarrow 4+\sqrt{3}=x$
${\overrightarrow{v}}_r=-3\hat{i}+4\hat{j}$
$tan\theta =\frac{2}{4}\Rightarrow \theta ={37}^o$