A man holds an umbrella at ${30}^{\circ }$ with the vertical to keep himself dry. He then runs at a speed of $10\text{m/s}$ 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

The correct option is C $10\sqrt{3}\text{m/s}$, $20\text{m/s}$
$v_m=$ velocity of man,
$v_{re}=$ velocity of rain w.r.t earth
$v_{rm}=$ velocity of rain w.r.t man

Given: Velocity of man, $v_m=10\text{m/s}$
Initially, the rain is falling at ${30}^{\circ }$ to the vertical


Using $\sin {30}^{\circ }=\frac{v_m}{v_{re}}$,
$v_{re}=\frac{v_m}{\sin {30}^{\circ }}=\frac{10}{1/2}=20\text{m/s}$
Again, $\cos {30}^{\circ }=\frac{v_{rm}}{v_{re}}$
or $v_{rm}=v_{re}\cos {30}^{\circ }$
$\Rightarrow v_{rm}=20\times \frac{\sqrt{3}}{2}=10\sqrt{3}\text{m/s}$