Rain is falling vertically downward with velocity `4m//s.` A man is moving horizontally with velocity `3 m//s`, the velocity of rain with respect to man is

Options

A. `5m//s` at an angle `tan^(-1)(4//3)` with horizontal
B. `5 m//s` at an angle `tan^(-1)(3//4)` with vertical
C. `5m//s` at an angle `tan^(-1)(4//3)` with vertical
D. Both `(1)` and `(2)`

Explanation

To find the velocity of rain with respect to the man, we can follow these steps:

### Step 1: Understand the given velocities
- The rain is falling vertically downward with a velocity of \(4 \, \text{m/s}\).
- The man is moving horizontally with a velocity of \(3 \, \text{m/s}\).

### Step 2: Represent the velocities in vector form
- The velocity of the rain can be represented as:
\[ \vec{V}_{\text{rain}} = 0 \hat{i} - 4 \hat{j} \, \text{m/s} \quad (\text{downward direction}) \]
- The velocity of the man can be represented as:
\[ \vec{V}_{\text{man}} = 3 \hat{i} + 0 \hat{j} \, \text{m/s} \quad (\text{horizontal direction}) \]

### 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:
\[ \vec{V}_{\text{rain, man}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{man}} \]
- Substituting the values:
\[ \vec{V}_{\text{rain, man}} = (0 \hat{i} - 4 \hat{j}) - (3 \hat{i} + 0 \hat{j}) \]
\[ = -3 \hat{i} - 4 \hat{j} \, \text{m/s} \]

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

### Step 5: Find the direction of the relative velocity
- The direction can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-4}{-3} = \frac{4}{3} \]
- Therefore, the angle \( \theta \) with respect to the horizontal can be calculated as:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

### Final Result
- The velocity of the rain with respect to the man is:
\[ 5 \, \text{m/s} \text{ at an angle of } \tan^{-1}\left(\frac{4}{3}\right) \text{ with the horizontal.} \]