Question

A man is standing on an inclined plane of inclination $\theta$ with the horizontal and rain is falling vertically with respect to the man. Now the man starts walking along the incline. Mark the correct option.

• A. If man walks up the incline, rain may appear to come horizontally
• B. If man walks down the incline, rain may appear to come horizontally
• C. No matter whether man walks up or down, rain can never appear to come horizontally
• D. The rain always appears to come vertically

Answer 1

Answer: (B)

If man walks down the incline, rain may appear to come horizontally

Answer 2

To solve this problem, we need to use the concept of relative velocity. Let $\vec{v}_r$ be the velocity of rain with respect to the ground. Let $\vec{v}_m$ be the velocity of the man with respect to the ground. The velocity of rain with respect to the man is given by $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m$.

Step 1: Determine the absolute velocity of rain.

The problem states that when the man is standing on the inclined plane, "rain is falling vertically with respect to the man". When the man is standing, his velocity with respect to the ground is $\vec{v}_m = 0$. Therefore, the velocity of rain with respect to the man is $\vec{v}_{rm} = \vec{v}_r - 0 = \vec{v}_r$. Since this relative velocity is described as "falling vertically", it means the absolute velocity of rain is purely vertical and downwards. Let's set up a coordinate system where the x-axis is horizontal and the y-axis is vertical (upwards). So, $\vec{v}_r = -v_r \hat{j}$, where $v_r$ is the speed of the rain.

Step 2: Analyze the case when the man walks up the incline.

The incline makes an angle $\theta$ with the horizontal. If the man walks up the incline with a speed $u$, his velocity vector $\vec{v}_m$ will have horizontal and vertical components: $\vec{v}_m = u \cos\theta \hat{i} + u \sin\theta \hat{j}$. Now, calculate the velocity of rain with respect to the man: $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (-v_r \hat{j}) - (u \cos\theta \hat{i} + u \sin\theta \hat{j})$ $\vec{v}_{rm} = -u \cos\theta \hat{i} - (v_r + u \sin\theta) \hat{j}$. For the rain to appear to come horizontally, its vertical component with respect to the man must be zero. The vertical component of $\vec{v}_{rm}$ is $V_{rm,y} = -(v_r + u \sin\theta)$. Since $v_r > 0$, $u > 0$ (as the man is walking), and $\sin\theta > 0$ (for an inclined plane, $0 < \theta < \pi/2$), the term $(v_r + u \sin\theta)$ is always positive. Therefore, $V_{rm,y}$ will always be negative and can never be zero. This means if the man walks up the incline, rain can never appear to come horizontally. Option A is incorrect.

Step 3: Analyze the case when the man walks down the incline.

If the man walks down the incline with a speed $u$, his velocity vector $\vec{v}_m$ will have horizontal and vertical components: $\vec{v}_m = u \cos\theta \hat{i} - u \sin\theta \hat{j}$. Now, calculate the velocity of rain with respect to the man: $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (-v_r \hat{j}) - (u \cos\theta \hat{i} - u \sin\theta \hat{j})$ $\vec{v}_{rm} = -u \cos\theta \hat{i} + (-v_r + u \sin\theta) \hat{j}$. For the rain to appear to come horizontally, its vertical component with respect to the man must be zero. The vertical component of $\vec{v}_{rm}$ is $V_{rm,y} = -v_r + u \sin\theta$. We need to check if $V_{rm,y} = 0$: $-v_r + u \sin\theta = 0$ $u \sin\theta = v_r$ $u = \frac{v_r}{\sin\theta}$. Since $v_r > 0$ and $\sin\theta > 0$ (for an inclined plane), there exists a positive speed $u$ for which this condition is satisfied. If the man walks down the incline with this specific speed $u = \frac{v_r}{\sin\theta}$, the vertical component of the rain's velocity with respect to the man becomes zero, and the rain appears to come horizontally. The horizontal component would be $-u \cos\theta = -\frac{v_r}{\sin\theta} \cos\theta = -v_r \cot\theta$. So, option B is correct.

Step 4: Evaluate other options.

Option C states that rain can never appear to come horizontally, which contradicts our finding in Step 3. So, C is incorrect. Option D states that rain always appears to come vertically. This is only true when the man is standing still. When he starts walking, the relative velocity changes. So, D is incorrect.