A convex mirror is placed on the ceiling at the intersection...

A convex mirror is placed on the ceiling at the intersection of two hallways. If a person stands directly underneath the mirror, the person's shoe is a distance of $195 \mathrm{cm}$ from the mirror. The mirror forms an image of the shoe that appears $12.8 \mathrm{cm}$ behind the mirror's surface. What is the mirror's focal length? What is the magnification of the image? Is the image real or virtual? Is the image upright or inverted?

Key Concepts

Mirror Equation

The mirror equation, given by 1/p + 1/q = 1/f, relates the object distance (p), image distance (q), and focal length (f) of a mirror. It is fundamental for analyzing how images are formed, allowing one to compute any one of these variables when the other two are known. Proper use of this equation requires careful attention to sign conventions.

Sign Conventions in Mirror Problems

In problems involving mirrors, sign conventions are essential to correctly assign positive or negative values to distances and focal lengths. Typically, object distances are taken as positive when the object is in front of the mirror, image distances are negative for virtual images (which form behind the mirror), and convex mirrors have a negative focal length. These conventions ensure consistency in calculations.

Convex Mirror Properties

Convex mirrors are curved such that they always produce images that are virtual, upright, and smaller than the object. Regardless of the object’s position, the image appears to be formed behind the mirror, a result of the diverging nature of the reflected light rays. This makes convex mirrors widely useful in applications like safety mirrors.

Magnification

Magnification is defined as the ratio of the image height to the object height, and it can also be expressed in terms of distances as m = -q/p. A positive magnification indicates that the image is upright compared to the object, while a negative magnification would indicate an inverted image. In the context of convex mirrors, the images are typically reduced and upright.

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