A man running on a horizontal road at 8 kmph finds the rain is falling vertically. He increases the speed to 12 kmph and find that the drops make angle ${30}^0$  with the vertical               

Let velocity of rain $V_r=V_{r_x}\widehat{i}+V_{r_y}\widehat{j}$

 Velocity of rain wrt man, $V_{rm}=\left(V_{r_x}-V_m\right)\widehat{i}+V_{r_y}\widehat{j}$

Initially, the man observes rain to be vertical when running at 8 kmph. 

Therefore $V_{r_x}-V_m=0\Rightarrow V_{r_x}=8 kmph$

When the man increases his speed to 12 kmph horizontally, 

$\tan 30=\frac{V_{r_x}-V_m}{V_{r_y}}=\frac{8-12}{V_{r_y}}$

$V_{r_y}=-4\sqrt{3}kmph$

$V_r=8\widehat{i}-4\sqrt{3}\widehat{j}=4\sqrt{2^2+3}=\sqrt{112}kmph$