Problem 34 - College Physics

Question

A convex mirror is on the exterior of the passenger side of many trucks (see Conceptional Question \(8 \mathrm{a}\) ). If the focal length of such a mirror is \(-40.0 \mathrm{~cm}\), what will be the location and height of the image of a car that is \(2.0 \mathrm{~m}\) high and (a) \(100 \mathrm{~m}\) and (b) \(10.0 \mathrm{~m}\) behind the truck mirror?

Step-by-Step Solution

Answer

For a) image is at -39.84 cm and 0.798 cm high; for b) image is at -36.36 cm and 7.272 cm high.

Step 1: Understand the Mirror Formula

The mirror formula relates the object distance \(u\), the image distance \(v\), and the focal length \(f\) using the equation: \[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\]. For convex mirrors, the focal length \(f\) is negative. We have \(f = -40.0\, \text{cm}\).

Step 2: Convert Units

Convert the object distances from meters to centimeters because the focal length is given in cm. Thus, for part (a), the object distance \(u = 100\, \text{m} = 10000\, \text{cm}\). For part (b), \(u = 10\, \text{m} = 1000\, \text{cm}\).

Step 3: Use Mirror Formula for Part (a)

Substitute \(f = -40.0\, \text{cm}\) and \(u = 10000\, \text{cm}\) into the mirror formula: \[\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-40} - \frac{1}{10000}\]. Calculate the value of \(v\).

Step 4: Calculate Image Distance for Part (a)

Perform the calculation: \[\frac{1}{v} = \frac{1}{-40} - \frac{1}{10000}\]. Solve for \(v\) to find \(v \approx -39.84\, \text{cm}\). This negative sign indicates the image is a virtual image located behind the mirror.

Step 5: Use Magnification Formula for Height in Part (a)

The magnification \(m\) is given by \(m = \frac{h'}{h} = -\frac{v}{u}\). For part (a), substitute \(v = -39.84\, \text{cm}\) and \(u = 10000\, \text{cm}\) to find the image height \(h'\).

Step 6: Calculate Image Height for Part (a)

Given \(h = 200\, \text{cm}\), use \(m = -\frac{v}{u} = \frac{h'}{h}\) to find \(h'\): \[h' = h \times \frac{v}{u} = 200 \times \frac{-39.84}{10000} \approx -0.798\, \text{cm}\].

Step 7: Repeat Mirror Formula for Part (b)

Substitute \(f = -40.0\, \text{cm}\) and \(u = 1000\, \text{cm}\) into the mirror formula for part (b): \[\frac{1}{v} = \frac{1}{-40} - \frac{1}{1000}\]. Calculate the value of \(v\).

Step 8: Calculate Image Distance for Part (b)

Perform the calculation: \[\frac{1}{v} = \frac{1}{-40} - \frac{1}{1000}\]. Solve for \(v\) to find \(v \approx -36.36\, \text{cm}\). This again indicates a virtual image.

Step 9: Use Magnification Formula for Height in Part (b)

For part (b), with \(v = -36.36\, \text{cm}\) and \(u = 1000\, \text{cm}\), substitute in \(m = -\frac{v}{u}\) to find \(h'\).

Step 10: Calculate Image Height for Part (b)

Given \(h = 200\, \text{cm}\), \use \(m = -\frac{v}{u} = \frac{h'}{h}\) to find \(h'\): \[h' = 200 \times \frac{-36.36}{1000} \approx -7.272\, \text{cm}\].

Key Concepts

Image Formation

In optics, image formation in convex mirrors is quite unique. Unlike concave mirrors, which can produce real images, convex mirrors always form virtual images. A virtual image is one that appears to be behind the mirror rather than in front of it. This happens because the reflected light rays diverge upon reflection, meaning they spread out. When your eyes perceive these diverging rays, they extrapolate the rays backward to where they seem to converge behind the mirror, forming a virtual image. These images are also upright and smaller than the actual object. These characteristics make convex mirrors particularly useful in applications requiring a wide field of view, like passenger-side mirrors on vehicles.

Focal Length

The focal length of a mirror is a measure of how strongly it converges or diverges light rays. In convex mirrors, the focal length is negative. This is due to the fact that the focal point, the point where light rays parallel to the principal axis appear to originate from, lies on the opposite side of the reflecting surface. In the context of our exercise, the focal length of the convex mirror is \(-40 \, \text{cm}\). Having a negative focal length is consistent with the nature of convex mirrors and is crucial for calculations like finding the location of the virtual image. Understanding focal length helps in applying the mirror formula effectively.

Mirror Formula

The mirror formula is a fundamental equation in optics, linking the object distance (\(u\)), the image distance (\(v\)), and the focal length (\(f\)) of a mirror. It is mathematically expressed as \(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\). For convex mirrors, it's vital to remember that the focal length is negative, which affects the interpretation and results of this formula. When using the mirror formula:

• Ensure distances are converted to consistent units, like centimeters.
• Signs are essential - in our case, a negative focal length and a negative image distance indicate a virtual image.
• By calculating \(v\), you determine where the virtual image appears relative to the mirror.

Magnification Formula

The magnification formula gives us insight into how the size of the image compares to the size of the object. It's given by \( m = \frac{h'}{h} = -\frac{v}{u}\), where \(m\) is the magnification, \(h'\) is the image height, \(h\) is the object height, \(v\) is the image distance, and \(u\) is the object distance.

• A positive magnification indicates that the image is upright, while a negative magnification suggests an inverted image.
• In the case of convex mirrors, the image is always upright and smaller than the object, hence the magnification is positive but less than 1.
• This formula helps us understand not just the size alteration but also the orientation of the image formed by the mirror.

Understanding the magnification process allows us to predict how an object will be perceived through the mirror.