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)`

Answer

To find the velocity of the raindrops as they fall on a man walking in the rain, we can use vector addition. Here’s a step-by-step solution:

### Step 1: Identify the velocities
- The velocity of the rain (Vr) is given as 3 km/h vertically downward.
- The velocity of the man (Vm) is given as 4 km/h horizontally.

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

### Step 3: Calculate the relative velocity of the rain with respect to the man
- The relative velocity of the rain with respect to the man (Vrm) 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
- To find the magnitude of the relative velocity, we can use the Pythagorean theorem:
\[ |\vec{V_{rm}}| = \sqrt{(-4)^2 + (-3)^2} \]
- Calculating this gives:
\[ |\vec{V_{rm}}| = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km/h} \]

### Conclusion
The raindrops will fall on the man with a velocity of **5 km/h**.

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