An observer moves towards a plane mirror with a speed of `2(m)/(sec)`. The speed of the image with respect to the observer is

Options

A. `1(m)/(sec)`
B. `2(m)/(sec)`
C. `4(m)/(sec)`
D. `8(m)/(sec)`

Explanation

To solve the problem of finding the speed of the image with respect to the observer moving towards a plane mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- An observer is moving towards a plane mirror with a speed of \( v_o = 2 \, \text{m/s} \).
- The image formed by the plane mirror appears to be at the same distance behind the mirror as the observer is in front of it.

2. **Determine the Speed of the Image**:
- The speed of the image \( v_i \) with respect to the mirror is equal to the speed of the observer but in the opposite direction. Therefore, the speed of the image is also \( 2 \, \text{m/s} \) but directed away from the mirror.
- Thus, we can express the speed of the image as \( v_i = -2 \, \text{m/s} \) (negative sign indicates the opposite direction).

3. **Calculate the Speed of the Image with Respect to the Observer**:
- The speed of the image with respect to the observer is given by the formula:
\[ v_{io} = v_i - v_o \]
- Substituting the values:
\[ v_{io} = (-2 \, \text{m/s}) - (2 \, \text{m/s}) = -4 \, \text{m/s} \]

4. **Interpret the Result**:
- The negative sign indicates that the image is moving towards the observer. The magnitude of the speed is \( 4 \, \text{m/s} \).

5. **Final Answer**:
- The speed of the image with respect to the observer is \( 4 \, \text{m/s} \).