Question

Rain, driven by the wind, falls on a railway compartment with a velocity of $20m{s}^{-1}$, at an angle of ${30}^{\circ }$ to the vertical. The train moves, along the direction of wind flow, at a speed of $108km{h}^{-1}$. Determine the apparent velocity of rain for a person sitting in the train.

• A. $20\sqrt{7}m{s}^{-1}$
• B. $10\sqrt{7}m{s}^{-1}$
• C. $15\sqrt{7}m{s}^{-1}$
• D. $10\sqrt{7}km{h}^{-1}$

Answer 1

The correct option is B $10\sqrt{7}m{s}^{-1}$

Speed of train $=108\times \frac{5}{18}=30m{s}^{-1}$
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 $v_{R,T}={\overrightarrow{v}}_R-{\overrightarrow{v}}_T\Rightarrow {\overrightarrow{v}}_R+(-{\overrightarrow{v}}_T)$
Let $\overrightarrow{OR}$ and $\overrightarrow{RT}$ represent the vectors, respectively, in magnitude and direction.
$O{T}^2=O{R}^2+R{T}^2=2OR\cdot RT\cos {120}^{\circ }$
$={20}^2+{30}^2-2\times 20\times 30\times \frac{1}{2}$
$=400+900-600=700=\sqrt{700}m{s}^{-1}$
$=10\sqrt{7}m{s}^{-1}$.