Question

Rain, driven by the wind, falls on a railway compartment with a velocity of $20m/s$, at an angle of ${30}^{\circ }$ to the vertical. The train moves, along the direction of wind flow, at a speed of $108km/hr$. Determine the apparent velocity of rain for a person sitting in the train.

• A. $20\sqrt{7}m/s$
• B. $15\sqrt{7}m/s$
• C. $12\sqrt{7}m/s$
• D. $10\sqrt{7}m/s$

Answer 1

The correct option is D $10\sqrt{7}m/s$

Speed of train $=108\times \frac{5}{18}=30m/s$
Let ${\overrightarrow{v}}_R$ and ${\overrightarrow{v}}_T$ represent the respective velocities of rain and train.
Now, the relative velocity of rain w.r.t. person (train) is given by ${\overrightarrow{v}}_{RT}=\overrightarrow{v_R}-\overrightarrow{v_T}$
$={\overrightarrow{v}}_R+(-{\overrightarrow{v}}_T)$
Let $OR$ and $RT$ represent the velocity vectors.


$O{T}^2=O{R}^2+R{T}^2+2OR\times RT\cos {120}^{\circ }$
$={20}^2+{30}^2-2\times 20\times 30\times \frac{1}{2}$
$=400+900-600=700m/s$

$OT={\overrightarrow{v}}_{RT}=10\sqrt{7}m/s$
Hence the correct option is (d).