Problem 69 - University Physics with Modern Physics

Question

If you run away from a plane mirror at \(3.60 \mathrm{m} / \mathrm{s},\) at what speed does your image move away from you?

Step-by-Step Solution

Answer

The image moves away at 7.20 m/s.

Step 1: Understanding the Situation

When you stand in front of a plane mirror and see your reflection, the image appears to be the same distance behind the mirror as you are in front of it. Therefore, any movement you make away from the mirror, your reflection will mimic by moving away from the mirror at the same speed, doubling the overall distance between you and your image.

Step 2: Determining the Relative Speed

If you move away from the mirror at a speed of \(3.60 \mathrm{m/s}\), your image also moves away from the mirror at \(3.60 \mathrm{m/s}\). Since both you and your image are moving away from each other, the relative speed between you and your image is the sum of both speeds.

Step 3: Calculating the Speed of the Image

The speed at which you and your image move apart is the sum of your speed and the image's speed: \(3.60 \mathrm{m/s} + 3.60 \mathrm{m/s} = 7.20 \mathrm{m/s}\). Therefore, your image moves away from you at \(7.20 \mathrm{m/s}\).

Key Concepts

Understanding Relative Speed

Relative speed is a fundamental concept in physics, especially when analyzing movements involving reflections, such as those in a plane mirror. When you move towards or away from a mirror, your image behaves as if it were a separate entity doing the exact opposite. This is key to understanding relative speed between you and your image.

Consider this: if you move away from a mirror at a speed of 3.60 m/s, your reflection also moves away from the mirror at the same speed. The overall distance between you and your image thus increases faster, because both you and your image are moving in opposite directions.

• Your speed: 3.60 m/s
• Image speed: 3.60 m/s
• Relative speed: 3.60 m/s + 3.60 m/s = 7.20 m/s

This doubling effect occurs because the mirror doesn't merely reflect your speed like a regular object; it reflects the concept of distance, causing both you and your image's movement to add up.

Principles of Image Formation

Image formation in plane mirrors is based on the principle that light reflects off surfaces at the same angle at which it arrives. This reflection creates an image behind the mirror that appears to mimic your actions perfectly but is laterally inverted.

When you look at yourself in a mirror, your brain perceives the image as being the same distance behind the mirror as you are in front of it. This symmetrical placement results from light traveling from you to the mirror and reflecting back to your eyes.

• Images appear the same size as the object.
• Distance to the mirror equals the distance from the mirror.
• The image is laterally inverted (left-right swapped).

Knowing how images form helps in comprehending how the distance doubles when both you and your reflection move away at the same speed. It's as if the mirror has a magical depth, positioning everything accurately behind it.

Basics of Motion Analysis in Reflections

Motion analysis in the context of reflections involves understanding how objects and their images move relative to each other and to their surroundings. Plane mirrors offer a unique scenario where both the object and its reflection seem to participate in the same motion although they don't actually exist beyond the mirror's surface.

When we analyze motion, we must account for observation points, such as how we perceive our movement and our image's movement. This helps determine apparent behaviors and relative motion.

• Observe the object's position and movement.
• Track the image's corresponding and mirrored motions.
• Evaluate how motions interact, such as how relative speed is resolved.

In exercises like moving away from a mirror, motion analysis lets us logically deduce that since both you and your image are moving away from a central point (the mirror), the cumulative effect is a relative speed greater than either of your speeds alone. Applying these principles simplifies predicting outcomes in mirrored environments.