Problem 5 - College Physics

Question

A small dog sits \(3.0 \mathrm{~m}\) in front of a plane mirror. (a) Where is the dog's image in relation to the mirror? (b) If the dog jumps at the mirror at a speed of \(1.0 \mathrm{~m} / \mathrm{s}\), how fast does the dog approach its image?

Step-by-Step Solution

Answer

(a) 3.0 m behind the mirror. (b) 2.0 m/s.

Step 1: Understanding Image Formation by a Plane Mirror

In a plane mirror, the image of an object appears to be the same distance behind the mirror as the object is in front of it. If the dog sits 3.0 m in front of the mirror, the image will also be 3.0 m behind the mirror.

Step 2: Calculate the Total Distance Between Dog and Its Image

The total distance between the dog and its image in the mirror is the sum of the distance from the dog to the mirror and the mirror to the image. This distance is \(3.0 \text{ m} + 3.0 \text{ m} = 6.0 \text{ m}\).

Step 3: Determine the Relative Speed of the Dog Towards Its Image

When the dog moves toward the mirror at a speed of \(1.0 \, \text{m/s}\), the image will approach the dog as if it moves at the same speed towards the mirror. Thus, their relative speeds add up, so the dog approaches its image at a speed of \(1.0 \, \text{m/s} + 1.0 \, \text{m/s} = 2.0 \, \text{m/s}\).

Key Concepts

Plane Mirror

A plane mirror is a flat reflective surface that reflects light to form images. When light rays hit a plane mirror, they bounce back according to the law of reflection. This means that the angle at which the light ray strikes the mirror, known as the angle of incidence, is equal to the angle at which it reflects off the mirror, called the angle of reflection.

Plane mirrors can be found in everyday life, such as in bathroom mirrors. They form images that are upright, laterally inverted (flipped sideways), and of the same size as the object. Importantly, for a plane mirror, the virtual image appears to be the same distance behind the mirror as the object is in front.

Image Formation

When an object is in front of a plane mirror, the light rays coming from the object reflect on the mirror to reach our eyes, forming an image. This image formation occurs such that the image is always formed at the same perpendicular distance behind the mirror, making it appear as if the object has a twin on the other side of the mirror.

In the case of the small dog sitting 3.0 meters in front of the mirror, its image will form 3.0 meters behind the mirror. Though it is virtual, meaning you cannot project it onto a screen, your eyes perceive the image directly in line with the object. This visible illusion works consistently across any plane mirror, allowing for a variety of practical applications, from simple reflections to complex imaging systems.

Relative Speed

Relative speed is the measure of how quickly one object is moving with respect to another. When two objects are moving towards one another, the relative speed is the sum of their individual speeds.

In our exercise, when the dog leaps towards the mirror at 1.0 m/s, its image, which appears to be moving in the opposite direction towards the dog at the same speed, creates a relative speed effect. This makes the dog approach its image at twice the speed, resulting in a relative speed of 2.0 m/s between the dog and its image.

Understanding the concept of relative speed is crucial for various applications in physics, including collision analysis and motion studies, where determining the effective speed between objects is necessary.

Mirror Image Distance

The mirror image distance refers to the virtual distance between the reflective surface and the perceived image in a plane mirror. This distance directly corresponds to the actual distance of the object from the mirror.

In practical terms, if an object is placed at a certain distance from a plane mirror, its image will seem to be the same distance behind the mirror. For example, if our dog is 3.0 meters away from the mirror, the image will also appear to be 3.0 meters behind, making the total perceived distance between the dog and its image 6.0 meters.

• This symmetrical property of image formation is a fundamental aspect of optics in plane mirrors.
• It helps in ensuring that decorations, designs, or other setups using mirrors maintain accuracy and balance.

Such insights into mirror image distances are essential for designing optics and understanding spatial relationships in mirrored environments.