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

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Q. 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 $10\, km / h$. He finds that raindrops are hitting his head vertically. The actual speed of raindrops is :

EAMCETEAMCET 2006

Solution:

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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}=10 \,km / h $
$v_{r, g} =\frac{10}{\sin 30^{\circ}}=20 \,km / h$