A car is moving on horizontal ground with a velocity vm `s^(-1)` with respect to earth. The driver of the car observes the rain to be falling vertically downwards with velocity 2vm `s^(-1)` . What is the velocity of the rain with respect to the driver ?

Options

A. 3v
B. `vsqrt(3)`
C. `vsqrt(5)`
D. none of these

Explanation

To solve the problem step by step, we need to analyze the situation involving the car, the rain, and their respective velocities.

### Step 1: Understand the given information
- The car is moving horizontally with a velocity \( v_m \) (let's denote it as \( v \)).
- The rain is observed to be falling vertically downwards with a velocity of \( 2v_m \) (denote it as \( 2v \)) from the driver's perspective.

### Step 2: Establish the reference frames
- The velocity of the car with respect to the ground is \( \vec{V}_{cg} = v \hat{i} \) (where \( \hat{i} \) is the unit vector in the horizontal direction).
- The velocity of the rain with respect to the driver is \( \vec{V}_{rd} = 0 \hat{i} - 2v \hat{j} \) (where \( \hat{j} \) is the unit vector in the vertical direction).

### Step 3: Use the relative velocity concept
The velocity of the rain with respect to the ground can be expressed using the relative velocity formula:
\[ \vec{V}_{rg} = \vec{V}_{rd} + \vec{V}_{cg} \]
Where:
- \( \vec{V}_{rg} \) is the velocity of the rain with respect to the ground.
- \( \vec{V}_{rd} \) is the velocity of the rain with respect to the driver.
- \( \vec{V}_{cg} \) is the velocity of the car with respect to the ground.

### Step 4: Substitute the known values
Substituting the values we have:
\[ \vec{V}_{rg} = (0 \hat{i} - 2v \hat{j}) + (v \hat{i} + 0 \hat{j}) \]
This simplifies to:
\[ \vec{V}_{rg} = v \hat{i} - 2v \hat{j} \]

### Step 5: Calculate the magnitude of the rain's velocity with respect to the ground
To find the magnitude of the velocity of the rain with respect to the ground, we calculate:
\[ |\vec{V}_{rg}| = \sqrt{(v)^2 + (-2v)^2} \]
\[ |\vec{V}_{rg}| = \sqrt{v^2 + 4v^2} = \sqrt{5v^2} = v\sqrt{5} \]

### Step 6: Conclusion
Thus, the velocity of the rain with respect to the driver is \( v\sqrt{5} \) m/s.