Question

A boy is running on a levelled road with velocity v with a long hollow tube in his hand. Water is falling vertically downwards with velocity u. at what angle to the vertical should we incline the tube so that the water drops enters without touching its side.

Answer 1

The rain even though falling perpendicularly would appear slanted to the boy. We can make a right-angled triangle using the velocity of boy and rain. The angle opposite to the boy's velocity would be the angle to be made with the vertical.

Complete step by step solution:
When we look at the question clearly, we can understand that there is no wind flowing. Thus, from our common sense we might be forced to think that the rain would fall down straight.as a matter of fact we are right, the rain is falling down perpendicular to the ground as there is no wind involved.
But velocity is a relative term, and hence for the running boy the rain would appear to be slanting or tilted by an angle $\theta$ with the vertical.
To calculate the value of this angle we can use the help of basic geometry and trigonometry,
As we know the rain is falling perpendicularly with a velocity $u$ and the boy is running horizontally(perpendicular to the rain) with a velocity $v$. These velocities form a right-angled triangle, with the angle opposite to the horizontal being $\theta$.
Now, using trigonometry,
$\tan \theta =\dfrac{v}{u}$
$\therefore \theta ={{\tan }^{-1}}\dfrac{v}{u}$

So the tube can be held at an angle equal to ${\tan ^{ - 1}}\dfrac{v}{u}$ so that the rain droplets enter the pipe, without touching the sides.

Note:
The angle should not be greater than or less than the given value. If a wind blows, the condition changes and we need to add the velocity of wind to the velocity of rain. Also, then it wouldn’t necessarily form a right-angled triangle with the horizontal.