E1.3 Relative Velocity (Rain falling vertically downward):

When a man is stationary and rain is falling vertically downward, the man has to held the umbrella vertically to save himself from the rain. The relative velocity of rain with respect to the man is in the vertical direction.
 

Suppose rain is falling vertically downward with velocity ($v_r$) and the man is walking in the horizontal direction with velocity ($v_m$), as shown in figure. 

The raindrops fall on him in the resultant direction with resultant velocity. This resultant velocity is inclined with vertical. 

Mathematically, 

$tan\theta = \frac{v_m}{v_r}$ 

Thus, a man walking in rain should hold his umbrella making an angle $\theta$ with the vertical, such that:

$\theta = tan^{-1}\frac{v_m}{v_r}$

To understand this concept is by directly solving the problem. 

A man is going due east with a velocity of $5 km/hr$. Rain falls vertically downwards at a speed of $10 km/hr$. Calculate the angle at which he should hold his umbrella so as to save himself from the rain? 

Answer: $26.56^0$  [Hence, the man has to hold his umbrella $26.56^0$ east of vertical.]

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