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 9.09 m.

Step 1: Understanding the Scenario

In this exercise, we are considering how images are reflected in a plane mirror. A camera is positioned 4.30 m in front of the mirror, and a hummingbird sits at the same level, 5.00 m to the right of the camera, and 3.30 m in front of the mirror. We need to find where the bird's image appears in the mirror and the distance from the camera to this image.

Step 2: Calculating Bird's Image Position

The bird is located 3.30 m from the mirror. In a plane mirror, the image appears as far behind the mirror as the object is in front of it. Hence, the image of the bird will also be 3.30 m behind the mirror. Additionally, the position to the right remains consistent, thus the image is 5.00 m to the right as well.

Step 3: Determining the Total Distance

The camera is 4.30 m in front of the mirror, so the total distance between the camera and where the image of the bird appears behind the mirror is the sum of the camera's and bird's position from the mirror: 4.30 m in front plus 3.30 m behind the mirror equals 7.60 m as a straight line in the depth direction.

Step 4: Calculating the Diagonal Distance

To find the direct line distance between the camera and the bird's image position, we consider this as the diagonal of a right triangle, with the horizontal distance 5.00 m and the total front-back distance as 7.60 m. Using the Pythagorean theorem: \[\text{Distance} = \sqrt{(7.60 \, \text{m})^2 + (5.00 \, \text{m})^2}.\]

Step 5: Final Calculation

Calculate the diagonal distance: \[7.60^2 = 57.76 \, \text{m}^2, \5.00^2 = 25.00 \, \text{m}^2, \\text{Distance} = \sqrt{57.76 + 25.00} = \sqrt{82.76}.\]Calculating this gives approximately 9.09 m.

Key Concepts

Image Formation

When an object is placed in front of a plane mirror, an image is formed by reflecting light rays. The image appears to be located behind the mirror at the same distance as the object is in front. For instance, if a hummingbird is 3.30 m in front of a mirror, its image will appear 3.30 m behind the mirror. This happens because a plane mirror reverse backwards the path of the light but doesn't change directions left or right. Hence, the image maintains the same lateral position as the object. In this scenario, since the bird is also horizontally 5.00 m to the right of the camera, its image will likewise appear 5.00 m to the right, just behind the mirror. This gives a comprehensive view of image formation in plane mirrors, allowing us to visualize where images appear based on object positions.

Reflection

Reflection is a fundamental principle that occurs when light waves bounce off a surface instead of being absorbed. In the context of plane mirrors, reflection obeys two main laws: the angle of incidence equals the angle of reflection, and both angles are measured relative to the normal, a line perpendicular to the reflective surface. This perfect reflection characteristic of plane mirrors causes the light rays to follow predictable paths, creating virtual images. A virtual image is one that cannot be projected on a screen because the light rays don't actually meet but appear to diverge from a common point behind the mirror. This consistent behavior of light reflects clarity in understanding image location, using geometry to determine positions and distances.

Geometric Optics

Geometric optics deals with light propagation in terms of rays and is pivotal in explaining phenomena like reflection and refraction. When tackling problems like the image of a hummingbird in a mirror, understanding geometric optics is essential. Concepts from geometry, such as the Pythagorean theorem, help calculate exact distances. In our case, the bird's direct image positions form a right triangle when combined with the camera's location. With a horizontal separation of 5.00 m and combined depth of 7.60 m from front to back, the direct distance to the bird's image is found using the equation:

• Calculate the square of each side of the triangle
• Add these squares together
• Find the square root of the sum, yielding 9.09 m

Thus, geometric optics not only clarifies image position but enables precise computation of such relative distances.