Problem 1 - Fundamentals of Physics

Question

You look through a camera toward an image of a hummingbird in a plane mirror. The camera is \(4.30 \mathrm{~m}\) in front of the mirror. The bird is at camera level, \(5.00 \mathrm{~m}\) to your right and \(3.30 \mathrm{~m}\) from the mirror. What is the distance between the camera and the apparent position of the bird's image in the mirror?

Step-by-Step Solution

Answer

The distance is approximately 9.10 m.

Step 1: Analyze the setup

First, we need to understand the situation. We have a camera placed in front of a plane mirror and a hummingbird positioned right of the camera. The mirror reflects the image of the hummingbird, creating an apparent position of the image on the opposite side of the mirror.

Step 2: Calculate the apparent position of the bird's image

In a plane mirror, the image appears as far behind the mirror as the object is in front of it. The bird is 3.30 m from the mirror, so its image will also be 3.30 m behind the mirror.

Step 3: Determine horizontal distance between camera and bird

The bird is 5 meters to the right of the camera. This is the same for the image, which will also appear 5 meters to the right of the camera's line of sight.

Step 4: Calculate the total distance

The camera is 4.30 m in front of the mirror, and the bird's image is 3.30 m behind the mirror. Therefore, the total distance the image is behind the camera is the sum of these two distances: 4.30 m + 3.30 m = 7.60 m. This distance forms a triangle with the 5.00 m horizontal distance we found in Step 3.

Step 5: Use Pythagoras' theorem

To find the straight-line distance between the camera and the image of the bird in the mirror, we apply the Pythagorean theorem: \[\sqrt{(7.60)^2 + (5.00)^2} = \sqrt{57.76 + 25} = \sqrt{82.76} \approx 9.10 \text{ m} \] This is the distance sought.

Key Concepts

Plane Mirrors

Understanding how plane mirrors work is a fundamental concept in optics. Plane mirrors are flat, reflective surfaces that produce images by reflecting light. In essence, when light rays hit a plane mirror, they bounce off at the same angle but in an opposite direction.

Key characteristics of plane mirrors include:

• Images are virtual, meaning they cannot be projected on a screen and appear to be behind the mirror.
• Images are the same size as the object being reflected.
• Images are laterally inverted, which means the left and right sides are switched.

In our exercise, the hummingbird is the object, and its reflected image appears as if it is the same distance from the mirror as the bird is in front of it. This fundamental property of plane mirrors helps us determine the apparent position of the bird's image.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical principle used frequently in physics problem solving, especially when dealing with distances and angles. It states that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side, known as the hypotenuse.

Mathematically, it is expressed as:

• \[ a^2 + b^2 = c^2 \]

In this exercise, we applied the Pythagorean Theorem to determine the straight-line distance from the camera to the image of the bird in the mirror. The side measured directly in front of the mirror (combined camera and image distance) and the side measuring horizontally from mirror to camera create a right-angled triangle. Solve for the hypotenuse to get the distance between the camera and the image.

Optics

Optics is the branch of physics that deals with the behavior and properties of light, including its interactions with matter. In the context of our problem, the key focus is on reflection—a primary phenomenon studied in optics.

Reflection occurs when light bounces off a surface. With a plane mirror, light reflects in a clear and predictable manner, allowing objects to appear as images behind the mirror surface. Understanding optics principles like reflection and refraction helps explain how images are formed and why they appear where they do.

When tackling optics problems, always consider:

• The nature of the reflective surface: flat (plane), curved, etc.
• Light's incident and reflected angles: For plane mirrors, these angles are equal.
• The characteristics of the image formed: orientation, size, and location.

Image Formation in Mirrors

The formation of images in mirrors can be fascinating. In plane mirrors, image formation follows a simple rule: images appear as if they are located the same distance behind the mirror as the object is in front of it. This distance is called the "apparent position." The characteristics of images formed in mirrors include:

• Virtual images, meaning they cannot be captured on a physical screen.
• Images are the same size as the object and upright.
• Lateral inversion, where the left side of the object appears as the right side in the image.

Understanding these properties is crucial when solving problems involving reflections. You need to visualize where the image would appear based on the distance from the mirror and the relative positions of objects and viewers. In our hummingbird exercise, these rules help us derive the apparent distance, thereby using geometry to find the direct distance from the camera to the image.