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

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$20 \sqrt{7} {ms}^{- 1}$

$10 \sqrt{7} {ms}^{- 1}$

$15 \sqrt{7} {ms}^{- 1}$

$10 \sqrt{7} {kmh}^{- 1}$

answer is B.

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Detailed Solution

$Speed of train = 108 \times \frac{5}{18} = 30 {ms}^{- 1}$

Let ${\vec{v}}_{R} and {\vec{v}}_{T}$ represent the respective velocities of rain and train.

Now, the relative velocity of rain w.r.t. person (train) is given by $v_{R, T} = {\vec{v}}_{R} - {\vec{v}}_{T} \Rightarrow {\vec{v}}_{R} + (- {\vec{v}}_{T})$

Let $\vec{O R} and \vec{R T}$ represent the vectors, respectively, in magnitude and direction.

$\begin{array}{l} O T^{2} = O R^{2} + R T^{2} + 2 O R \cdot R T \cos 120^{\circ} \\ = 20^{2} + 30^{2} - 2 \times 20 \times 30 \times \frac{1}{2} \\ = 400 + 900 - 600 = 700 = \sqrt{700} {ms}^{- 1} \\ = 10 \sqrt{7} {ms}^{- 1} \end{array}$