Question

A man holds an umbrella at ${30}^{\circ }$ with the vertical to keep himself dry. He, then, runs at a speed of $10{\text{ms}}^{-1}$ and finds the rain drops to be hitting vertically. Speed of the rain drops w.r.t. the running man and w.r.t. earth are

• A. $20{\text{ms}}^{-1},10{\text{ms}}^{-1}$
• B. $10{\text{ms}}^{-1},20\sqrt{3}{\text{ms}}^{-1}$
• C. $10\sqrt{3}{\text{ms}}^{-1},20{\text{ms}}^{-1}$
• D. $20{\text{ms}}^{-1},10\sqrt{3}{\text{ms}}^{-1}$

Answer 1

The correct option is C $10\sqrt{3}{\text{ms}}^{-1},20{\text{ms}}^{-1}$

$v_{\text{m}}=$ velocity of man
$v_{\text{re}}=$ velocity of rain w.r.t. earth
$v_{\text{rm}}=$ velocity of rain w.r.t. man
Velocity of man $|{\overrightarrow{v}}_{\text{m}}|=10m{s}^{-1}$
Using $\text{sin}{30}^{\circ }=\frac{v_{\text{m}}}{v_{\text{re}}}$
or $v_{\text{re}}=\frac{v_{\text{m}}}{\text{sin}30}=\frac{10}{\frac{1}{2}}=20m{s}^{-1}$
Again $\text{cos}{30}^{\circ }=\frac{v_{\text{rm}}}{v_{\text{re}}}$
or $v_{\text{rm}}=v_{\text{re}}\text{cos}30=20\times \frac{\sqrt{3}}{2}=10\sqrt{3}m{s}^{-1}$