Rain is falling vertically with a speed of 40 $m s^{-1}$. Wind starts blowing with a speed of 16 $m s^{-1}$ in the west to east direction. How should a person, who is standing, hold his umbrella to avoid getting wet?
- At an angle of about 22° with vertical towards east
- At an angle of about 22° with vertical towards west
- At an angle of about 66° with vertical towards east
- At an angle of about 66° with vertical towards west
The Correct Option is B
Solution and Explanation
To determine how a person should hold their umbrella, let's analyze the situation considering both the rain and wind speeds as vector components.
- Rain is falling vertically, so it has a vertical speed, \(v_r = 40 \, \text{m/s}\).
- Wind blows horizontally from west to east, hence it has a horizontal speed, \(v_w = 16 \, \text{m/s}\).
- The resultant velocity of the rain considering both components (vertical fall and horizontal wind) can be calculated using vector addition. This resultant will determine the direction in which the person should hold the umbrella.
To find the angle \(\theta\) with respect to the vertical, use the tangent function:
\(\tan \theta = \frac{v_w}{v_r}\)
Substitute the given values:
\(\tan \theta = \frac{16}{40} = 0.4\)
Now calculate \(\theta\):
\(\theta = \tan^{-1}(0.4) \approx 21.8^\circ\)
Therefore, the person should hold the umbrella at an angle of approximately \(22^\circ\) with the vertical. As the wind is blowing from west to east, the umbrella should be tilted towards the west to counteract the effect of the wind.
Hence, the correct answer is:
At an angle of about 22° with vertical towards west
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