Problem 74 - University Physics with Modern Physics

Question

Rear-View Mirror. A mirror on the passenger side of your car is convex and has a radius of curvature with magnitude 18.0 \(\mathrm{cm} .\) (a) Another car is behind your car, 9.00 m from the mirror, and this car is viewed in the mirror by your passenger. If this car is 1.5 \(\mathrm{m}\) tall, what is the height of the image? (b) The mirror has a warning attached that objects viewed in it are closer than they appear. Why is this so?

Step-by-Step Solution

Answer

(a) The image height is approximately 1.49 cm. (b) Objects appear further away because convex mirrors diminish image size, creating the perception of increased distance.

Step 1: Understanding the Mirror Formula

The mirror formula relates the object distance (\( u \)) and image distance (\( v \)) to the focal length (\( f \)) of the mirror. For a mirror, this is given by \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \). For a convex mirror, the focal length \( f \) is positive and related to the radius of curvature \( R \) by \( f = \frac{R}{2} \).

Step 2: Calculate the Focal Length

Given the radius of curvature \( R = 18.0 \text{ cm} \), calculate the focal length \( f \). Since the mirror is convex, the focal length is positive. \[ f = \frac{18 \text{ cm}}{2} = 9 \text{ cm} \].

Step 3: Apply the Mirror Formula

Convert the object distance from meters to centimeters: \( u = 900 \text{ cm} \). Using the mirror formula, substitute \( f = 9 \text{ cm} \) and \( u = -900 \text{ cm} \) (negative since the object is in front of the mirror): \[ \frac{1}{9} = \frac{1}{v} - \frac{1}{900} \].

Step 4: Solve for Image Distance

Rearrange the equation to solve for \( v \):\[ \frac{1}{v} = \frac{1}{9} + \frac{1}{900} \]. Calculate the right-hand side: \[ \frac{1}{v} = \frac{900 + 9}{8100} = \frac{909}{8100} \]. Now, invert to find \( v \): \[ v = \frac{8100}{909} \approx 8.91 \text{ cm} \].

Step 5: Use Magnification to Find Image Height

The magnification \( m \) for a mirror is given by \( m = \frac{h'}{h} = \frac{v}{u} \), where \( h' \) is the image height and \( h = 1.5 \text{ m} = 150 \text{ cm} \). Calculate \( m \):\[ m = \frac{8.91}{-900} \approx -0.0099 \]. Hence, \( h' = m \times 150 \text{ cm} \approx -1.485 \text{ cm} \).

Step 6: Explanation of Warning

Convex mirrors produce smaller images of the objects due to their diverging nature, which makes objects appear further away than their actual distance. Because the mirror shrinks the image, it gives the impression that the object is further than it actually is.

Key Concepts

Mirror Formula

The mirror formula is a fundamental concept in optics that connects the distances involved when using mirrors. It's given by the equation \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \). Here's what each symbol means:

• \( f \): Focal length of the mirror.
• \( v \): Image distance from the mirror.
• \( u \): Object distance from the mirror.

For a convex mirror, the focal length \( f \) is always considered positive. This is because convex mirrors spread the light rays apart, rather than focusing them. The focal length relates to the radius of curvature with the formula \( f = \frac{R}{2} \). By knowing these relations, we can determine where an image will form and its characteristics. Understanding how to manipulate this formula is key to solving mirror-related problems.

Image Height

In the context of mirrors, the image height is calculated using the concept of magnification. Magnification (\( m \)) shows how a mirror enlarges or reduces the size of an image relative to the actual object. It's calculated using the formula \( m = \frac{h'}{h} = \frac{v}{u} \), where:

• \( h' \): The height of the image.
• \( h \): The actual height of the object.
• \( v \): The image distance from the mirror (calculated using the mirror formula).
• \( u \): The object distance from the mirror.

For a convex mirror, magnification is always less than one, indicating that images are smaller than the object. Additionally, the negative sign in magnification implies that the image formed is virtual and upright compared to the object. This shrinkage explains why objects in such mirrors appear smaller and farther away than they truly are.

Focal Length

The focal length of a mirror is an essential parameter in optics that dictates how light interacts with the mirror. Specifically, for a convex mirror, the light rays spread out, giving the mirror a positive focal length. It's determined using the relationship \( f = \frac{R}{2} \), where \( R \) is the radius of curvature of the mirror. For example, if a convex mirror has a radius of curvature of 18.0 cm, its focal length would be 9.0 cm (i.e., \( f = \frac{18}{2} = 9 \) cm). This property is critical for understanding and applying the mirror formula, as it assists in calculating where and how an image will form. Knowing the focal length not only helps when using the mirror formula but also illustrates the nature of images formed by convex mirrors, which tend to be virtual and reduced in size.