There are two plane mirror with reflecting suface faciing each other.The mirrors are moving with speed v away form each other .A point object is placed between the mirrors.The velocity of the n-th image will be

Options

A. nv
B. 2nv
C. 3nv
D. 4nv

Explanation

To find the velocity of the n-th image formed by two plane mirrors moving away from each other with a speed \( v \), we can follow these steps:

### Step 1: Understand the setup
We have two plane mirrors facing each other, and they are moving away from each other with a speed \( v \). A point object is placed between the mirrors.

### Step 2: Analyze the motion of the mirrors
Since the mirrors are moving away from each other, we need to consider the relative motion of the images formed by the mirrors. The object is stationary, and the mirrors are moving.

### Step 3: Determine the velocity of the first image
The first image of the object will be formed by the first mirror. As the mirror moves away from the object with speed \( v \), the image will appear to move towards the object with the same speed \( v \). Thus, the velocity of the first image \( v_1 \) is:
\[ v_1 = v \]

### Step 4: Determine the position of the second image
The second image is formed by the reflection of the first image in the second mirror. Since the second mirror is also moving away from the first image with speed \( v \), the second image will appear to move away from the first image with speed \( v \) as well. Therefore, the velocity of the second image \( v_2 \) is:
\[ v_2 = v + v = 2v \]

### Step 5: Generalize for the n-th image
Continuing this pattern, we can see that:
- The first image moves at speed \( v \).
- The second image moves at speed \( 2v \).
- The third image, which is the reflection of the second image in the first mirror, will move at speed \( 3v \).
- The n-th image will thus have a velocity of \( n \times v \).

### Step 6: Final expression for the n-th image
The velocity of the n-th image \( v_n \) can be expressed as:
\[ v_n = 2nv \]
This accounts for the fact that each subsequent image reflects the motion of the previous image and the mirrors.

### Conclusion
Thus, the velocity of the n-th image is given by:
\[ \text{Velocity of the n-th image} = 2nv \]

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