A man walking downhill with velocity $V_0$ finds that his umbrella gives him maximum protection from rain when he holds it such that the stick is perpendicular to the hill surface. When the man turns back and climbs the hill with velocity $V_0$, he finds that it is most appropriate to hold the umbrella stick vertically. Then the actual speed of raindrops in terms of $V_0$ is: (the inclination of the hill is $\theta =37°$).

The correct option is B

$\frac{\sqrt{73}}{3}{V}_0$

Step 1: Given

Velocity while walking downwards: $V_0$

Velocity while walking upwards: $V_0$

Inclination angle: $\theta =37°$

Velocity component of rain along x-axis is $V_x$

Velocity component of rain along y-axis is $V_y$

Velocity of rain is $V_r$

Velocity of man is $V_m$

Relative velocity of rain with respect to man is $V_{rm}$

Step 2: Formula Used

The resultant velocity of rain is $V=\sqrt{{V_x}^2+{V_y}^2}$, where the velocity component of rain along the x-axis is $V_x$ and the velocity component of rain along the y-axis is $V_y$.

Step 3: Find the x-component of the velocity of rain

Calculate the velocity of rain relative to man while going downhill by subtracting the man's velocity from that of the rain.

$$\begin{gathered} \overrightarrow{V_{rm}}=\overrightarrow{V_r}-\overrightarrow{V_m} \\ =\left(V_x\hat{i}+V_y\hat{j}\right)-V_0\hat{i} \\ =\left(V_x-V_0\right)\hat{i}+\hat{j} \end{gathered}$$

Calculate the x component of the velocity of rain by equating $V_x-V_0$ to 0, since the relative velocity has no x component.

$$\begin{gathered} V_x-V_0=0 \\ \Rightarrow V_x=V_0 \end{gathered}$$

Step 4: Find the y-component of the velocity of rain

Calculate the velocity of rain relative to man while going uphill by adding the man's velocity to that of the rain.

$$\begin{gathered} \overrightarrow{V_{rm}}=\overrightarrow{V_r}+\overrightarrow{V_m} \\ =\left(V_x\hat{i}+V_y\hat{j}\right)+V_0\hat{i} \\ =\left(V_x+V_0\right)\hat{i}+V_y\hat{j} \\ =\left(V_0+V_0\right)\hat{i}+V_y\hat{j} \\ =2V_0\hat{i}+V_y\hat{j} \end{gathered}$$

Calculate the y component of the velocity of rain by using the tan formula for the inclined angle

$$\begin{gathered} \tan \theta =\frac{2V_0}{V_y} \\ \Rightarrow \frac{3}{4}=\frac{2V_0}{V_y} \\ \Rightarrow V_y=\frac{4}{3}\times 2V_0 \\ \Rightarrow V_y=\frac{4}{3}{V}_0 \end{gathered}$$

Step 3: Calculate the velocity of rain using the formula

$$\begin{gathered} V=\sqrt{{V_x}^2+{V_y}^2} \\ =\sqrt{{\left(V_0\right)}^2+{\left(\frac{8}{3}{V}_0\right)}^2} \\ =\sqrt{\frac{73}{9}{V}_0} \\ =\frac{\sqrt{73}}{3}{V}_0 \end{gathered}$$

Hence, Option (B) is correct. the velocity of rain is $\frac{\sqrt{73}}{3}{V}_0$.