Question

A car travels with a speed of 40 km/h. Rain drops fall vertically but the traces on the side window make an angle of \( 30^\circ \) with the vertical. Find the magnitude of rain’s velocity with respect to the car.

• A. \( 40\sqrt{3} \) km/h
• B. \( \frac{40}{\sqrt{3}} \) km/h
• C. 80 km/h
• D. \( \frac{80}{\sqrt{3}} \) km/h

Hint:

Use vector resolution and relative velocity to calculate motion under angles.

Answer 1

The Correct Option is D

Let rain’s velocity with respect to the car be \( v_r \).
Let horizontal component = car's speed = 40 km/h, and angle with vertical is \( \theta = 30^\circ \).
Using \( \tan \theta = \frac{\text{horizontal}}{\text{vertical}} \Rightarrow \tan 30^\circ = \frac{40}{v_v} \Rightarrow \frac{1}{\sqrt{3}} = \frac{40}{v_v} \Rightarrow v_v = 40\sqrt{3} \)
Now, actual rain velocity = \( \sqrt{40^2 + (40\sqrt{3})^2} = \sqrt{1600 + 4800} = \sqrt{6400} = 80 \) km/h.
So with respect to car: magnitude of relative velocity = \( \frac{80}{\sqrt{3}} \)