A man runs towards plane mirror at a speed of 15 m/sec. What is the speed of his image?

Options

A. 7.5 m/s
B. 15m/s
C. 30m/s
D. 45m/s

Answer

To solve the problem of finding the speed of the man's image in a plane mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A man is running towards a plane mirror at a speed of 15 m/s.
- The image of the man in the mirror will be formed behind the mirror at the same distance as the man is in front of it.

2. **Distance Calculation**:
- Let’s assume the distance of the man from the mirror is \( x \) meters.
- The image of the man will be at a distance \( x \) behind the mirror.

3. **Movement of the Man**:
- As the man runs towards the mirror at 15 m/s, after 1 second, he will have covered a distance of 15 meters.
- Therefore, his new distance from the mirror will be \( x - 15 \) meters.

4. **Position of the Image**:
- The image will also move closer to the mirror as the man approaches it.
- After 1 second, the image will be at a distance of \( x - 15 \) behind the mirror.

5. **Total Distance Between Man and Image**:
- Initially, the distance between the man and his image was \( 2x \) (the distance in front of the mirror plus the distance behind the mirror).
- After 1 second, the distance between the man and his image becomes:
\[ 2(x - 15) = 2x - 30 \text{ meters} \]

6. **Calculating the Change in Distance**:
- The change in distance between the man and his image over 1 second is:
\[ (2x) - (2x - 30) = 30 \text{ meters} \]

7. **Speed of the Image**:
- The speed of the image can be calculated as the distance traveled by the image in 1 second. Since the distance between the man and his image decreased by 30 meters in 1 second, the speed of the image is:
\[ \text{Speed of the image} = \frac{\text{Distance traveled}}{\text{Time taken}} = \frac{30 \text{ meters}}{1 \text{ second}} = 30 \text{ m/s} \]

### Final Answer:
The speed of the man's image is **30 m/s**.