Question

Rain is falling vertically with a speed of $20m/s$ relative to air. A person is running in the rain with a velocity of $5m/s$ and a wind is also blowing with a speed of $15m/s$ (both towards east). Find the tan of angle with the vertical at which the person should hold his umbrella so that he may not get drenched.

• A. $\frac{1}{2}$
• B. $2$
  
• C. $3$
  
• D. $6$
  

Answer 1

The correct option is A $\frac{1}{2}$

Let East be along +ve x-axis and vertically up be along +ve y-axis

$\Rightarrow$ Speed of man$=\overrightarrow{V_m}=5\hat{i}$

$\Rightarrow$ Speed of wind$=\overrightarrow{V_w}=15\hat{i}$

$\Rightarrow$ Speed of rain relative to wind$=\overrightarrow{V_r}-\overrightarrow{V_w}=-20\hat{j}$

$\Rightarrow$ Speed of rain$=\overrightarrow{V_r}=15\hat{i}-20\hat{j}$

Now, speed of man relative to rain$=\overrightarrow{V_m}-\overrightarrow{V_r}=5\overrightarrow{i}-(15\hat{i}-20\hat{j})=-10\hat{i}+20\hat{j}$

Tan of angle from vertical at which the person should hold his umbrella$=10/20=1/2$

Hence, correct answer is option $A$