A man is walking on a road with a velocity 3kmhr. Suddenly rain starts falling. The velocity of rain is 10km/hr in vertically downward direction. the relative velocity of rain with respect to man is :-

Explanation

To solve the problem of finding the relative velocity of rain with respect to the man, we can follow these steps:

### Step 1: Understand the velocities involved
- The man is walking with a velocity of \( \vec{V_m} = 3 \, \text{km/hr} \) in the horizontal direction (let's assume to the right).
- The rain is falling vertically downward with a velocity of \( \vec{V_r} = 10 \, \text{km/hr} \).

### Step 2: Set up the coordinate system
- We will use a Cartesian coordinate system where:
- The x-axis represents the horizontal direction (to the right).
- The y-axis represents the vertical direction (downward).

### Step 3: Express the velocities in vector form
- The velocity of the man can be represented as:
\[ \vec{V_m} = 3 \, \hat{i} \, \text{km/hr} \]
- The velocity of the rain can be represented as:
\[ \vec{V_r} = -10 \, \hat{j} \, \text{km/hr} \]
(The negative sign indicates that the rain is falling downward, which is in the negative y-direction.)

### Step 4: 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 values:
\[ \vec{V_{rm}} = (-10 \, \hat{j}) - (3 \, \hat{i}) = -3 \, \hat{i} - 10 \, \hat{j} \, \text{km/hr} \]

### Step 5: Calculate the magnitude of the relative velocity
- The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V_{rm}}| = \sqrt{(-3)^2 + (-10)^2} \]
- Calculating the squares:
\[ |\vec{V_{rm}}| = \sqrt{9 + 100} = \sqrt{109} \]
- Therefore, the magnitude of the relative velocity is:
\[ |\vec{V_{rm}}| \approx 10.44 \, \text{km/hr} \]

### Final Answer
The relative velocity of rain with respect to the man is approximately \( 10.44 \, \text{km/hr} \) at an angle downward to the horizontal.

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