The writing on the passenger-side mirror of your car says...

The writing on the passenger-side mirror of your car says "Warning! Objects in mirror are closer than they appear." There is no such warning on the driver's mirror. Consider a typical convex passenger-side mirror with a focal length of $-80 \mathrm{cm} .$ A $1.5-\mathrm{m}$ -tall cyclist on a bicycle is $25 \mathrm{m}$ from the mirror. You are $1.0 \mathrm{m}$ from the mirror, and suppose, for simplicity, that the mirror, you, and the cyclist all lie along a line. a. How far are you from the image of the cyclist? b. How far would you have been from the image if the mirror were flat? c. What is the image height? d. What would the image height have been if the mirror were flat? Why is there a label on the passenger-side mirror?

The writing on the passenger-side mirror of your car says "Warning! Objects in mirror are closer than they appear." There is no such warning on the driver's mirror. Consider a typical convex passenger-side mirror with a focal length of $-80 \mathrm{cm} .$ A $1.5-\mathrm{m}$ -tall cyclist on a bicycle is $25 \mathrm{m}$ from the mirror. You are $1.0 \mathrm{m}$ from the mirror, and suppose, for simplicity, that the mirror, you, and the cyclist all lie along
a line.
a. How far are you from the image of the cyclist?
b. How far would you have been from the image if the mirror were flat?
c. What is the image height?
d. What would the image height have been if the mirror were flat? Why is there a label on the passenger-side mirror?

Key Concepts

Perceptual Distortion and Safety Warning

The warning label on a car's passenger-side mirror addresses the discrepancy between the actual object distance and the perceived distance of objects in a convex mirror. Due to the mirror's curvature reducing the image size and changing the apparent distance, objects appear farther away than they really are, which is crucial for safe driving awareness.

Planar Mirror Reflection

Planar, or flat, mirrors produce images that are the same size as the object and are located at the same distance behind the mirror as the object is in front. This ideal reflection is used as a baseline for comparison with curved mirrors, highlighting how curvature causes variations in apparent distance and size.

Magnification in Mirrors

Magnification in mirror optics is described by the ratio of the image height to the object height, which can also be expressed as the negative ratio of the image distance to the object distance. This concept is important for understanding how the size of the image compares to the size of the object and explains why images in convex mirrors appear reduced.

Convex Mirrors

Convex mirrors curve outward and always produce virtual, diminished, and upright images regardless of the object's distance from the mirror. They have a negative focal length and are widely used in applications like vehicle side mirrors because they provide a wider field of view, even though they distort depth perception.

Sign Conventions

Sign conventions are critical in mirror calculations, especially when dealing with convex mirrors that have negative focal lengths. They determine the positive or negative values for object distances, image distances, and focal lengths, ensuring that virtual images (which appear behind the mirror) are correctly indicated and that the mirror equation correctly reflects the physics of mirror imaging.

Mirror Equation

The mirror equation, given by 1/object distance + 1/image distance = 1/focal length, is a fundamental relationship in geometric optics used to determine the position of an image formed by a mirror. It allows us to relate the object distance, image distance, and the focal length of the mirror, and is key to solving problems involving mirror image formation.

Transcript

00:01 Okay, so in this situation we have a mirror, but we're going to treat this mirror as a lens.

00:09 And there's our image plane here.

00:14 And we know that there's a cyclist behind our mirror that's 1 .5 meters tall and 25 meters away.

00:23 And we ourselves are one meter away.

00:27 And the lens has a focal length of negative 0 .8 meters.

00:35 Okay, so for part a, we want to know where the image of the cyclist is.

00:44 Where is the image of the cyclist relative to the person? okay, so that's going to be our standard 1 over f equals 1 over do plus 1 over d .i, which is going to become 1 over negative 0 .8 equals distance the object.

01:05 It's cyclist 25 meters plus 1 over d .i.

01:13 And d .i is going to come out to be 0 .77 or negative 0 .77.

01:20 So i drew this out as a lens just to keep this clear, but really it's a mirror.

01:25 So how it would look is here's a mirror, here's the driver right here, one meter away.

01:34 And then the driver's going to look at the mirror.

01:36 And then the image, the cyclist is back here, right? looks at the mirror and then reflects back.

01:43 But i think this is a little bit easier of a way to visualize it.

01:47 Okay.

01:48 And, okay, so to answer the question how far we are, we are.

01:56 Okay, so the cyclist is going to be negative 0 .77 away from the mirror.

02:01 So that would actually be in this diagram on this side.

02:07 But if there's a mirror, they'd appear to be.