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Rain is falling vertically with a speed of $35m{s^{ - 1}}$. Winds start blowing after some time with a speed of $12m{s^{ - 1}}$ in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella?

Last updated date: 23rd Feb 2024
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Hint: In order to find the direction in which the boy should hold the umbrella, first of all we need to find the resultant of the rain and wind. Then we can easily find the direction in which the umbrella has to be held.

Complete step by step answer:
The speed of rain in the question is given as, ${v_r} = 35m{s^{ - 1}}$
And the speed of the wind is given as, ${v_w} = 12m{s^{ - 1}}$

We know that the resultant of the rain and the wind will be the vector sum of the rain and wind.
So, we can find the magnitude of the resultant as, $v = \sqrt {{v_r} + {v_w}}$
$\Rightarrow v = \sqrt {{{35}^2} + {{12}^2}}$
$\Rightarrow v = \sqrt {1225 + 144}$
$\Rightarrow v = \sqrt {1369} = 37m{s^{ - 1}}$
Therefore, the resultant of the rain and wind is $37m{s^{ - 1}}$.

Now we need to find the direction of the resultant rain and wind.
We know that the direction of the resultant can be found by using the relation,
$\tan \theta = \dfrac{{{v_w}}}{{{v_r}}}$
So, the direction will be,$\theta = {\tan ^{ - 1}}\dfrac{{12}}{{35}}$
$\therefore \theta = {18.92^\circ }$
In order to be safe from the rain, one has to hold the umbrella in the opposite direction of the resultant.

Hence, the boy needs to hold the umbrella at ${18.92^\circ }$ in the east direction.

Note: In the given question both the velocity vectors are perpendicular to each other. So, for the resultant of the velocity vectors we used $v = \sqrt {{v_r} + {v_w}}$. The boy needs to hold the umbrella in the opposite direction of the resultant to save him from the rain. Also, a resultant is the combination of two or more vectors. The quantities which have both magnitude and direction are known as vectors.