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 :-

Options

A. `sqrt(13) km//hr`
B. `sqrt(7)km//hr`
C. `sqrt(109) km//hr`
D. 13 km/hr.

Answer

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

### Step 1: Define the velocities
- The velocity of the man (Vm) is given as 3 km/hr in the horizontal direction. We can represent this as:
\[ \vec{V_m} = 3 \hat{i} \text{ km/hr} \]
- The velocity of the rain (Vr) is given as 10 km/hr in the vertical downward direction. We can represent this as:
\[ \vec{V_r} = -10 \hat{j} \text{ km/hr} \]

### Step 2: Calculate the relative velocity
The relative velocity of the rain with respect to the man (Vr_m) is calculated using the formula:
\[ \vec{V_{r/m}} = \vec{V_r} - \vec{V_m} \]
Substituting the values:
\[ \vec{V_{r/m}} = (-10 \hat{j}) - (3 \hat{i}) = -3 \hat{i} - 10 \hat{j} \text{ km/hr} \]

### Step 3: Find the magnitude of the relative velocity
To find the magnitude of the relative velocity vector, we use the Pythagorean theorem:
\[ |\vec{V_{r/m}}| = \sqrt{(-3)^2 + (-10)^2} \]
Calculating this gives:
\[ |\vec{V_{r/m}}| = \sqrt{9 + 100} = \sqrt{109} \text{ km/hr} \]

### Step 4: Conclusion
The magnitude of the relative velocity of rain with respect to the man is:
\[ |\vec{V_{r/m}}| = \sqrt{109} \text{ km/hr} \]

### Final Answer
The relative velocity of rain with respect to the man is \(\sqrt{109} \text{ km/hr}\).

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