Question

A man running on a horizontal road at $8{\text{ms}}^{-1}$ finds rain falling vertically. If he increases his speed to $12{\text{ms}}^{-1}$, he finds that drops make ${30}^{\circ }$ angle with the vertical. Find the velocity of rain with respect to the road.

• A. $4\sqrt{7}{\text{ms}}^{-1}$
• B. $8\sqrt{2}{\text{ms}}^{-1}$
• C. $7\sqrt{3}{\text{ms}}^{-1}$
• D. $8{\text{ms}}^{-1}$

Answer 1

The correct option is A $4\sqrt{7}{\text{ms}}^{-1}$

Let velocity of rain $V_r=V_{r_x}\hat{i}+V_{r_y}\hat{j}$
Velocity of rain wrt man,
$V_{rm}=(V_{r_x}-V_m)\hat{i}+V_{r_y}\hat{j}$

case (i)
$V_{r_x}-V_m=0\Rightarrow v_{r_x}=8{\text{ms}}^{-1}$
case (ii)
$\tan 30=\frac{V_{r_x}-V_m}{V_{r_y}}=\frac{8-12}{V_{r_y}}$
$V_{r_y}=-4\sqrt{3}{\text{ms}}^{-1}$

$V_r=8\hat{i}-4\sqrt{3}\hat{j}=4\sqrt{2^2+3}=4\sqrt{7}{\text{ms}}^{-1}$