A man holds an umbrella at 30° with the vertical to keep himself dry. He, then, runs at a speed of 10 ms^-1, and find the rain drops to be hitting vertically. Study the following statements and find the correct options.
i. Velocity of rain w.r.t. Earth is 20 m s^-1
ii. Velocity of rain w.r.t. man is $10 \sqrt{3} {ms}^{- 1}$
iii. Velocity of rain w.r.t. Earth is $30 {ms}^{- 1}$
iv. Velocity of rain w.r.t. man is $10 \sqrt{2} {ms}^{- 1}$

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Statements (i) and (ii) are correct.

Statements (i) and (iii) are correct.

Statements (iii) and (iv) are correct.

Statements (ii) and (iv) are correct.

answer is A.

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Detailed Solution

$\begin{array}{l} {\vec{v}}_{m} = Velocity of man \\ {\vec{v}}_{re} = Velocity of rain w.r.t. earth \\ {\vec{v}}_{rm} = Velocity of rain w.r.t. man \end{array}$
$man |{\vec{v}}_{m}| = 10 {ms}^{- 1}$
using $\sin 30^{\circ} = \frac{v_{m}}{v_{re}}$
$\begin{array}{l} v_{re} = \frac{v_{m}}{\sin 30^{\circ}} = \frac{10}{1 / 2} = 20 {ms}^{- 1} \\ \cos 30^{\circ} = \frac{v_{rm}}{v_{re}} \\ v_{rm} = v_{re} \cos 30 = \frac{20 \times \sqrt{3}}{2} = 10 \sqrt{3} {ms}^{- 1} \end{array}$