A man is running with a speed of 4 m/s, if he feels that rain drops are hitting him vertically with speed 3 m/s (along unit vector `-hatj`) then what can be the velocity of rain:

Options

A. `3hati-4hatj`
B. `4hati+3hatj`
C. `-3hati-4hatj`
D. `-4hati-3hatj`

Explanation

To solve the problem, we need to analyze the situation using the concept of relative velocity. Let's break it down step by step.

### Step 1: Understand the Given Information
- The man is running with a speed of \(4 \, \text{m/s}\).
- He perceives the rain to be falling vertically downwards with a speed of \(3 \, \text{m/s}\) (along the unit vector \(-\hat{j}\)).

### Step 2: Define the Velocities
- Let the velocity of the man be represented as:
\[ \vec{V}_{\text{man}} = 4 \hat{i} \]
(assuming he is running in the positive x-direction).

- The velocity of the rain with respect to the man is given as:
\[ \vec{V}_{\text{rain, man}} = -3 \hat{j} \]

### Step 3: Use the Relative Velocity Equation
The relative velocity of rain with respect to the man can be expressed as:
\[ \vec{V}_{\text{rain, man}} = \vec{V}_{\text{rain}} - \vec{V}_{\text{man}} \]
From this, we can rearrange to find the velocity of the rain:
\[ \vec{V}_{\text{rain}} = \vec{V}_{\text{rain, man}} + \vec{V}_{\text{man}} \]

### Step 4: Substitute the Known Values
Substituting the known values into the equation:
\[ \vec{V}_{\text{rain}} = (-3 \hat{j}) + (4 \hat{i}) \]
This simplifies to:
\[ \vec{V}_{\text{rain}} = 4 \hat{i} - 3 \hat{j} \]

### Step 5: Conclusion
Thus, the velocity of the rain is:
\[ \vec{V}_{\text{rain}} = 4 \hat{i} - 3 \hat{j} \]