Rain drops are falling vertically downward with a velocity 4kmph. The velocity with which they appear to fall down to a man travelling at 2 kmph horizontal is

Options

A. `sqrt(12)` kmph
B. `2sqrt(5)` kmph
C. 5 kmph
D. 2 kmph

Answer

To solve the problem, we need to find the apparent velocity of the raindrops as observed by a man moving horizontally. Here’s a step-by-step solution:

### Step 1: Understand the velocities involved
- The raindrops are falling vertically downward with a velocity \( V_r = 4 \, \text{km/h} \).
- The man is moving horizontally with a velocity \( V_m = 2 \, \text{km/h} \).

### Step 2: Set up the vector components
- The velocity of the rain can be represented as a vector pointing downward (negative y-direction).
- The velocity of the man can be represented as a vector pointing horizontally (positive x-direction).

### Step 3: Use the Pythagorean theorem
Since the rain is falling vertically and the man is moving horizontally, we can treat these two velocities as perpendicular components of a right triangle. The apparent velocity of the rain with respect to the man can be calculated using the Pythagorean theorem:

\[ V_{rm} = \sqrt{V_r^2 + V_m^2} \]

### Step 4: Substitute the values
Now we substitute the given values into the equation:

\[ V_{rm} = \sqrt{(4 \, \text{km/h})^2 + (2 \, \text{km/h})^2} \]

\[ V_{rm} = \sqrt{16 + 4} = \sqrt{20} \]

### Step 5: Simplify the result
We can simplify \( \sqrt{20} \):

\[ V_{rm} = \sqrt{4 \times 5} = 2\sqrt{5} \, \text{km/h} \]

### Conclusion
The velocity with which the raindrops appear to fall down to the man is \( 2\sqrt{5} \, \text{km/h} \).

### Final Answer
The correct option is \( 2\sqrt{5} \, \text{km/h} \).

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