Question

problem 23.06
two plane mirrors, nearly parallel, are facing each other ( d_1 = 2.5 ) m apart as in (figure 1). you stand ( d_2 = 1.5 ) m away from one of these mirrors and look into it. you will see multiple images of yourself
figure
part c
how far away from you is the second image of yourself in the mirror in front of you?
express your answer to two significant figures and include the appropriate units.
( d_{i2} = ) value m
submit previous answers request answer
incorrect; try again; 2 attempts remaining
part d complete previous part(s)
part e

Explanation:

Step1: Analyze the first image in the front mirror

The distance from you to the front mirror is \( d_2 = 1.5 \, \text{m} \), so the first image in the front mirror is \( 1.5 \, \text{m} \) behind the front mirror, so the distance from you to this first image is \( 1.5 + 1.5 = 3.0 \, \text{m} \). But we need the second image.

Step2: Analyze the image formation in the two mirrors

The distance between the two mirrors is \( d_1 = 2.5 \, \text{m} \). The distance from you to the first mirror (front mirror) is \( d_2 = 1.5 \, \text{m} \), so the distance from you to the second mirror is \( d_1 - d_2 = 2.5 - 1.5 = 1.0 \, \text{m} \).

The first image in the second mirror (the mirror opposite to the front one) is \( 1.0 \, \text{m} \) behind the second mirror. Then, this image acts as an object for the front mirror. The distance from this object (first image in the second mirror) to the front mirror is \( d_1 + 1.0 = 2.5 + 1.0 = 3.5 \, \text{m} \)? Wait, no. Wait, the distance between the two mirrors is \( d_1 = 2.5 \, \text{m} \). The first image in the second mirror is \( 1.0 \, \text{m} \) behind the second mirror, so the distance from this image to the front mirror is \( d_1 + 1.0 = 2.5 + 1.0 = 3.5 \, \text{m} \)? No, wait, the second mirror is \( 2.5 \, \text{m} \) from the front mirror. The object (you) is \( 1.0 \, \text{m} \) from the second mirror, so the first image in the second mirror is \( 1.0 \, \text{m} \) behind the second mirror, so the distance from this image to the front mirror is \( 2.5 + 1.0 = 3.5 \, \text{m} \). Then, the image of this image in the front mirror (the second image in the front mirror) is \( 3.5 \, \text{m} \) behind the front mirror.

Now, the distance from you to this second image is the distance from you to the front mirror plus the distance from the front mirror to the second image. You are \( 1.5 \, \text{m} \) from the front mirror, and the second image is \( 3.5 \, \text{m} \) behind the front mirror, so the total distance is \( 1.5 + 3.5 = 5.0 \, \text{m} \)? Wait, no, maybe another approach.

Wait, the second image in the front mirror: let's think step by step.

1. First, your image in the front mirror (image 1): distance from you to image 1 is \( 2d_2 = 2\times1.5 = 3.0 \, \text{m} \).

2. Your image in the second mirror (image 2): distance from you to the second mirror is \( d_1 - d_2 = 2.5 - 1.5 = 1.0 \, \text{m} \), so image 2 is \( 1.0 \, \text{m} \) behind the second mirror, so distance from you to image 2 is \( 1.0 + (2.5 + 1.0) \)? No, wait, the distance from you to image 2: you are \( 1.0 \, \text{m} \) from the second mirror, so image 2 is \( 1.0 \, \text{m} \) behind the second mirror, so the distance from you to image 2 is \( 1.0 + 1.0 + 2.5 \)? Wait, no. Let's use the formula for multiple images in two parallel mirrors.

The first image in the front mirror (image 1): distance from you: \( 2d_2 = 3.0 \, \text{m} \).

The first image in the second mirror (image A): distance from you: you are \( d_1 - d_2 = 1.0 \, \text{m} \) from the second mirror, so image A is \( 1.0 \, \text{m} \) behind the second mirror, so the distance from you to image A is \( (d_1 - d_2) + d_1 + d_2 \)? No, the distance from you to the second mirror is \( 1.0 \, \text{m} \), so the distance from you to image A (in the second mirror) is \( 1.0 + 1.0 = 2.0 \, \text{m} \) plus the distance between the mirrors? Wait, no. The distance from you to the second mirror is \( 1.0 \, \text{m} \), so the image in the second mirror is \( 1.0 \, \text{m} \) behind the second mirror, so the distance from you to that image is \( 1.0 \, \text{m} \) (your distance to the second mirror) + \( 1.0 \, \text{m} \) (image behind the second mirror) + \( 2.5 \, \text{m} \) (distance between the mirrors)? No, that's not right. Wait, the two mirrors are \( 2.5 \, \text{m} \) apart. You are \( 1.5 \, \text{m} \) from mirror 1 (front mirror), so \( 2.5 - 1.5 = 1.0 \, \text{m} \) from mirror 2 (back mirror).

Image in mirror 2 (image X) is \( 1.0 \, \text{m} \) behind mirror 2. Now, this image X is an object for mirror 1. The distance from image X to mirror 1 is the distance between the mirrors plus the distance from image X to mirror 2, which is \( 2.5 + 1.0 = 3.5 \, \text{m} \). So the image of X in mirror 1 (which is the second image of yourself in mirror 1) is \( 3.5 \, \text{m} \) behind mirror 1.

Now, the distance from you to this second image is your distance to mirror 1 plus the distance from mirror 1 to the second image: \( 1.5 + 3.5 = 5.0 \, \text{m} \)? Wait, but let's check again.

Wait, another way: the first image in mirror 1: distance from you: \( 2 \times 1.5 = 3.0 \, \text{m} \).

The second image in mirror 1: this comes from the image formed in mirror 2, then reflected back to mirror 1. The distance from you to mirror 2 is \( 1.0 \, \text{m} \), so image in mirror 2 is \( 1.0 \, \text{m} \) behind mirror 2. Then, the distance from this image to mirror 1 is \( 2.5 + 1.0 = 3.5 \, \text{m} \), so the image in mirror 1 of this image is \( 3.5 \, \text{m} \) behind mirror 1. Then, the distance from you to this second image is \( 1.5 + 3.5 = 5.0 \, \text{m} \). Wait, but let's verify with the total path.

Alternatively, the number of times the light reflects: first reflection in mirror 1 (image 1), then reflection in mirror 2 (image of image 1 in mirror 2), then reflection in mirror 1 (image of that image, which is the second image in mirror 1). Wait, no, the second image in mirror 1 is formed by two reflections: first in mirror 2, then in mirror 1.

The distance traveled by the light for the second image: from you to mirror 2 (1.0 m), mirror 2 to mirror 1 (2.5 m), mirror 1 to the image (3.5 m)? No, the distance from you to the second image in mirror 1 is: you to mirror 1 (1.5 m) + mirror 1 to image (3.5 m) = 5.0 m. Wait, but let's check with the formula.

Wait, the distance between the two mirrors is \( D = 2.5 \, \text{m} \), your distance to mirror 1 is \( x = 1.5 \, \text{m} \), so your distance to mirror 2 is \( D - x = 1.0 \, \text{m} \).

The first image in mirror 1: distance from you: \( 2x = 3.0 \, \text{m} \).

The first image in mirror 2: distance from you: \( 2(D - x) = 2.0 \, \text{m} \).

The second image in mirror 1: this is the image of the first image in mirror 2, reflected back to mirror 1. The distance from the first image in mirror 2 to mirror 1 is \( D + (D - x) = 2D - x = 5.0 - 1.5 = 3.5 \, \text{m} \)? Wait, no, \( 2D - x = 5.0 - 1.5 = 3.5 \, \text{m} \). Then, the distance from you to this second image is \( x + (2D - x) = 2D = 5.0 \, \text{m} \)? Wait, \( x + (2D - x) = 2D \), which is \( 5.0 \, \text{m} \). Wait, that's a simpler way. So the second image in mirror 1: the distance from you is \( 2D - 2x + 2x \)? No, wait, let's think of the total path.

Wait, the second image in mirror 1 is formed by two reflections: first, you reflect in mirror 2 (distance from you to mirror 2 is \( D - x \), so image in mirror 2 is \( D - x \) behind mirror 2, so distance from mirror 1 to this image is \( D + (D - x) = 2D - x \). Then, the image in mirror 1 of this image is \( 2D - x \) behind mirror 1. Then, the distance from you to this image is \( x + (2D - x) = 2D \)? Wait, no, \( x + (2D - x) = 2D \), which is \( 5.0 \, \text{m} \). Wait, but \( 2D = 5.0 \, \text{m} \), but that seems off. Wait, no, let's take numbers. \( D = 2.5 \, \text{m} \), \( x = 1.5 \, \text{m} \).

First image in mirror 1: distance from you: \( 2x = 3.0 \, \text{m} \).

First image in mirror 2: distance from you: \( 2(D - x) = 2.0 \, \text{m} \).

Second image in mirror 1: this is the image of the first image in mirror 2, reflected in mirror 1. The first image in mirror 2 is \( D - x = 1.0 \, \text{m} \) behind mirror 2, so the distance from mirror 1 to this image is \( D + (D - x) = 2.5 + 1.0 = 3.5 \, \text{m} \). So the image in mirror 1 is \( 3.5 \, \text{m} \) behind mirror 1. Then, the distance from you to this image is \( x + 3.5 = 1.5 + 3.5 = 5.0 \, \text{m} \). Yes, that's correct. So the second image in the front mirror is \( 5.0 \, \text{m} \) away from you. Wait, but let's check with another approach.

Alternatively, the number of images: for two parallel mirrors, the images are formed at distances \( 2x, 2(D - x), 2D + 2x, 2D + 2(D - x), \dots \) from you? No, maybe not. Wait, the first image in mirror 1: \( 2x \) from you. The second image in mirror 1: this is the image formed by reflecting the image in mirror 2, which is \( 2(D - x) \) from you, but then reflecting back to mirror 1, so the distance is \( 2D - 2x + 2x = 2D \)? Wait, no, \( 2D - 2x + 2x = 2D \), which is \( 5.0 \, \text{m} \). So that's the same as before. So the second image in mirror 1 is \( 5.0 \, \text{m} \) from you.

Wait, let's confirm with the diagram. You are 1.5 m from mirror 1, 1.0 m from mirror 2. The first image in mirror 1: 1.5 m behind mirror 1, so 3.0 m from you. The first image in mirror 2: 1.0 m behind mirror 2, so the distance from you to this image is 1.0 m (to mirror 2) + 1.0 m (behind mirror 2) + 2.5 m (between mirrors) = 4.5 m? Wait, no, that's not right. Wait, the distance from you to the first image in mirror 2 is 2*(distance from you to mirror 2) = 2*1.0 = 2.0 m. Because the image is 1.0 m behind mirror 2, so the distance from you to mirror 2 is 1.0 m, so total distance is 1.0 + 1.0 = 2.0 m. Then, this image (first image in mirror 2) is an object for mirror 1. The distance from this object to mirror 1 is 2.5 m (distance between mirrors) + 1.0 m (behind mirror 2) = 3.5 m. So the image of this object in mirror 1 is 3.5 m behind mirror 1. Then, the distance from you to this image is 1.5 m (to mirror 1) + 3.5 m (behind mirror 1) = 5.0 m. Yes, that's correct. So the second image in mirror 1 is 5.0 m from you.

Answer:

\( 5.0 \, \text{m} \)