Rain is falling vertically with a velocity of 3`kmh^-1`. A man walks in the rain with a velocity of 4`kmh^-1`. The rain drops will fall on the man with a velocity of

Options

A. 5`kmh^-1`
B. 4`kmh^-1`
C. 3`kmh^-1`
D. 1`kmh^-1`

Explanation

To solve the problem of finding the velocity of the rain as it falls on the man walking in the rain, we can follow these steps:

### Step 1: Understand the velocities involved
- The rain is falling vertically downwards with a velocity of \( V_r = 3 \, \text{km/h} \).
- The man is walking horizontally with a velocity of \( V_m = 4 \, \text{km/h} \).

### Step 2: Represent the velocities as vectors
- The velocity of the rain can be represented as a vector:
\[ \vec{V_r} = 0 \, \hat{i} - 3 \, \hat{j} \, \text{(downward direction)} \]
- The velocity of the man can be represented as:
\[ \vec{V_m} = 4 \, \hat{i} + 0 \, \hat{j} \, \text{(horizontal direction)} \]

### Step 3: Calculate the relative velocity of rain with respect to the man
- The relative velocity of the rain with respect to the man is given by:
\[ \vec{V_{rm}} = \vec{V_r} - \vec{V_m} \]
- Substituting the vectors:
\[ \vec{V_{rm}} = (0 \, \hat{i} - 3 \, \hat{j}) - (4 \, \hat{i} + 0 \, \hat{j}) = -4 \, \hat{i} - 3 \, \hat{j} \]

### Step 4: Find the magnitude of the relative velocity
- The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V_{rm}}| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{km/h} \]

### Step 5: Conclusion
- Therefore, the raindrops will fall on the man with a velocity of \( 5 \, \text{km/h} \).