# Rain is falling vertically with a speed of $20\;ms^{-1}$. A person is running in rain (from east) with a velocity of $5 ms^{-1}$ and a wind is blowing with speed of $15ms ^{-1}$ (from west).The angle with the vertical at which the person should hold his umbrella is

$(a)\;\tan ^{-1}\bigg(\frac{1}{2}\bigg)\quad (b)\;\tan ^{-1}\bigg(\frac{1}{3}\bigg) \quad (c)\;\tan ^{-1}\bigg(\frac{4}{5}\bigg) \quad (d)\;\tan ^{-1}(3)$

Velocity of rain $=v_r=20 j$
Velocity of man $=v_m=5 i$
Velocity of wind $=15 i$
Velocity of rain with respect to man when wind is blowing is
$\quad=15 i-5i+20j$
$\quad=20j+10 i$
$\tan \theta=\large\frac{20}{10}$
$\qquad=2$ from the horizontal.
Hence the person should hold is umbrella at $\alpha$ (with vertical)
$\quad=\tan ^{-1}\bigg(\large\frac{1}{2}\bigg)$
Hence a is the correct answer

edited May 24, 2014