A man walks in rain with a velocity of 5 kmph. The rain drops strike at him at an angle of `45^(@)` with the horizontal. The downward velocity of the rain drops will be :

Options

A. 5 kph
B. 4 kph
C. 3 kph
D. 1 kph

Answer

To find the downward velocity of the rain drops, we can analyze the situation using vector components and the given information.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A man walks in the rain with a velocity of 5 km/h.
- The rain strikes him at an angle of 45 degrees with the horizontal.

2. **Setting Up the Velocity Components**:
- Let \( V_m \) be the velocity of the man, which is given as \( 5 \, \text{km/h} \) in the horizontal direction.
- Let \( V_r \) be the velocity of the rain, which we need to find. The rain's velocity can be broken down into horizontal and vertical components.

3. **Using the Angle of Impact**:
- The rain strikes the man at an angle of 45 degrees. This means that the horizontal component of the rain's velocity \( V_{rh} \) and the vertical component \( V_{rv} \) are equal:
\[ V_{rh} = V_{rv} \]

4. **Relative Velocity Concept**:
- The relative velocity of the rain with respect to the man \( V_{rm} \) can be expressed as:
\[ V_{rm} = V_r - V_m \]
- Since the rain strikes the man at a 45-degree angle, we can write:
\[ V_{rm} = V_{rh} \hat{i} + V_{rv} \hat{j} \]
- Here, \( V_{rh} \) is the horizontal component and \( V_{rv} \) is the vertical component of the rain's velocity.

5. **Equating Components**:
- Since \( V_{rh} = V_{rv} \) and the rain strikes at an angle of 45 degrees, we can set:
\[ V_{rh} = V_{rv} = V \]
- Therefore, we have:
\[ V_{rm} = V \hat{i} + V \hat{j} \]

6. **Finding the Horizontal Component**:
- The horizontal component of the rain's velocity \( V_{rh} \) is equal to the man's velocity:
\[ V_{rh} = V_m = 5 \, \text{km/h} \]
- Thus, we have:
\[ V = 5 \, \text{km/h} \]

7. **Finding the Vertical Component**:
- Since \( V_{rv} = V \), the vertical component of the rain's velocity is also:
\[ V_{rv} = 5 \, \text{km/h} \]

8. **Conclusion**:
- The downward velocity of the rain drops is:
\[ \text{Downward velocity of rain} = 5 \, \text{km/h} \]

### Final Answer:
The downward velocity of the rain drops is **5 km/h**.