Question

A man standing on a road has to hold his umbrella at $30^{\circ }$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10km/h$. He finds that raindrops are hitting his head vertically. The actual speed of raindrops is :

• A. 20 km / h
• B. $10\sqrt{3}km/h$
• C. $20\sqrt{3}km/h$
• D. 10 km / h

Answer 1

Answer: (A)

When the man is at rest with respect to the ground, the rain comes to him at an angle $30^{\circ }$ with the vertical. This is the direction of the velocity of raindrops with respect to the ground.
Here, ${\vec{v}}_{r,g}=$ velocity of the rain with respect to the ground
${\vec{v}}_{m,g}=$ velocity of the man with respect to the ground
and ${\vec{v}}_{r,m}=$ velocity of the rain with respect to the man.
We have, ${\vec{v}}_{r,g}+{\vec{v}}_{r,m}+{\vec{v}}_{m,g}$
Taking horizontal components, Eq. (i) gives
or $v_{r,g}\sin 30^{\circ }=v_{m,g}=10km/h$
$v_{r,g}=\frac{10}{\sin 30^{\circ }}=20km/h$