Question

Rain driven by the wind, falls on a railway compartment with a velocity of $20\text{m/s}$, at an angle of ${30}^{\circ }$ to the vertical. The train moves, along the direction of wind flow, at a speed of $108\text{km/hr}$. Determine the apparent velocity ( in $\text{m/s}$ upto two decimal places) of rain as observed by a person sitting in the train.( Given $\sqrt{7}=2.64$)

Answer 1

Speed of train $=108\times \frac{5}{18}=30\text{m/s}$
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 ${\overrightarrow{v}}_{R.T}=\overrightarrow{v_R}-\overrightarrow{v_T}\Rightarrow \overrightarrow{v_R}+(-\overrightarrow{V_T})$
Let $\overrightarrow{OR}$ and $\overrightarrow{RT}$ represent the vectors respectively in magnitude and direction.

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