To solve the problem of finding the distances of the first three images from the mirror \( M_2 \) when two plane mirrors are parallel to each other, we can follow these steps:
### Step 1: Understand the setup
We have two parallel mirrors, \( M_1 \) and \( M_2 \), with an object \( O \) placed between them. The distance between the two mirrors is given as 15 cm. The object \( O \) is 5 cm away from mirror \( M_2 \).
### Step 2: Determine the position of the object
Since the object is 5 cm away from mirror \( M_2 \), we can denote the position of the object as follows:
- Distance from \( M_2 \) to \( O \): \( d_{O,M_2} = 5 \, \text{cm} \)
### Step 3: Calculate the position of the first image \( I_1 \)
The first image \( I_1 \) formed by mirror \( M_2 \) will be at the same distance from \( M_2 \) as the object \( O \) but on the opposite side. Therefore:
- Distance from \( M_2 \) to \( I_1 \): \( d_{I_1,M_2} = d_{O,M_2} = 5 \, \text{cm} \)
### Step 4: Calculate the position of the second image \( I_2 \)
The second image \( I_2 \) is formed by the reflection of the first image \( I_1 \) in mirror \( M_1 \). The distance from \( M_1 \) to the first image \( I_1 \) is:
- Distance from \( M_1 \) to \( O \): \( 15 \, \text{cm} - 5 \, \text{cm} = 10 \, \text{cm} \)
Thus, the distance from \( M_1 \) to \( I_1 \) is 10 cm, and the distance from \( M_1 \) to \( I_2 \) will also be 10 cm:
- Distance from \( M_2 \) to \( I_2 \): \( 15 \, \text{cm} + 10 \, \text{cm} = 25 \, \text{cm} \)
### Step 5: Calculate the position of the third image \( I_3 \)
The third image \( I_3 \) is formed by the reflection of the second image \( I_2 \) in mirror \( M_2 \). The distance from \( M_2 \) to \( I_2 \) is 25 cm, so the distance from \( M_2 \) to \( I_3 \) will be:
- Distance from \( M_2 \) to \( I_3 \): \( 25 \, \text{cm} + 5 \, \text{cm} = 35 \, \text{cm} \)
### Final Distances
Thus, the distances of the first three images from mirror \( M_2 \) are:
1. \( I_1 \): 5 cm
2. \( I_2 \): 25 cm
3. \( I_3 \): 35 cm
### Summary
The distances of the first three images from mirror \( M_2 \) are:
- \( 5 \, \text{cm}, 25 \, \text{cm}, 35 \, \text{cm} \)
