A man is walking at 1.0 m/s directly towards a flat mirror. At what
speed is his separation from his image reducing?
If a mirror on a wall is only big enough for you to see yourself from
your head down to your waist, can you see your entire body by backing
up? Test this experimentally and come up with an explanation for your
observations, including a ray diagram.
Note that when you do the experiment, it's easy to confuse yourself if the
mirror is even a tiny bit off of vertical. One way to check yourself is to
artificially lower the top of the mirror by putting a piece of tape or a postit
note where it blocks your view of the top of your head. You can then
check whether you are able to see more of yourself both above and below
by backing up.
In this chapter we've only done examples of mirrors with hollowed-out
shapes (called concave mirrors). Now draw a ray diagram for a curved
mirror that has a bulging outward shape (called a convex mirror). (a) How
does the image's distance from the mirror compare with the actual object's
distance from the mirror? From this comparison, determine whether the
magnification is greater than or less than one. (b) Is the image real or
virtual? Could this mirror ever make the other type of image?
As discussed in question 3, there are two types of curved mirrors,
concave and convex. Make a list of all the possible combinations of types
of images (virtual or real) with types of mirrors (concave and convex).
(Not all of the four combinations are physically possible.) Now for each
one, use ray diagrams to determine whether increasing the distance of the
object from the mirror leads to an increase or a decrease in the distance of
the image from the mirror.
Some tips: To draw a ray diagram, you need two rays. For one of these,
pick the ray that comes straight along the mirror's axis, since its reflection
is easy to draw. After you draw the two rays and locate the image for the
original object position, pick a new object position that results in the same
type of image, and start a new ray diagram, in a different color of pen,
right on top of the first one. For the two new rays, pick the ones that just
happen to hit the mirror at the same two places; this makes it much easier
to get the result right without depending on extreme accuracy in your
ability to draw the reflected rays.
If the user of an astronomical telescope moves her head closer to or
farther away from the image she is looking at, does the magnification
change? Does the angular magnification change? Explain. (For simplicity,
assume that no eyepiece is being used.)
In figure (e) in section 2.3, only the image of my forehead was located
by drawing rays. Trace the figure accurately, and make a new set of rays
coming from my chin, and locate its image. To make it easier to judge the
angles accurately, draw rays from the chin that happen to hit the mirror at
the same points where the two rays from the forehead were shown hitting
it. By comparing the locations of the chin's image and the forehead's
image, verify that the image is actually upside-down, as shown in the
The figure shows four points where rays cross. Of these, which are
Here's a game my kids like to play. I sit next to a sunny window, and
the sun reflects from the glass on my watch, making a disk of light on the
wall or floor, which they pretend to chase as I move it around. Is the spot a
disk because that's the shape of the sun, or because it's the shape of my
watch? In other words, would a square watch make a square spot, or do we
just have a circular image of the circular sun, which will be circular no
Apply the equation M=di/do to the case of a flat mirror.
Use the method described in the text to derive the equation relating
object distance to image distance for the case of a virtual image produced
by a converging mirror.
(a) Make up a numerical example of a virtual image formed by a
converging mirror with a certain focal length, and determine the magnification.
(You will need the result of problem 2.) Make sure to choose values
of do and f that would actually produce a virtual image, not a real one.
Now change the location of the object a little bit and redetermine the
magnification, showing that it changes. At my local department store, the
cosmetics department sells mirrors advertised as giving a magnification of
5 times. How would you interpret this?
(b) Suppose a Newtonian telescope is being used for astronomical
observing. Assume for simplicity that no eyepiece is used, and assume a
value for the focal length of the mirror that would be reasonable for an
amateur instrument that is to fit in a closet. Is the angular magnification
different for objects at different distances? For example, you could consider
two planets, one of which is twice as far as the other.
(a) Find a case where the magnification of a curved mirror is infinite. Is
the angular magnification infinite from any realistic viewing position? (b)
Explain why an arbitrarily large magnification can't be achieved by having
a sufficiently small value of do.
The figure shows a device for constructing a realistic optical illusion.
Two mirrors of equal focal length are put against each other with their
silvered surfaces facing inward. A small object placed in the bottom of the
cavity will have its image projected in the air above. The way it works is
that the top mirror produces a virtual image, and the bottom mirror then
creates a real image of the virtual image. (a) Show that if the image is to be
positioned as shown, at the mouth of the cavity, then the focal length of
the mirrors is related to the dimension h via the equation
(b) Restate the equation in terms of a single variable x=h/f, and show that
there are two solutions for x. Which solution is physically consistent with
the assumptions of the calculation?
A hollowed-out surface that reflects sound waves can act just like an inbending
mirror. Suppose that, standing near such a surface, you are able to
find point where you can place your head so that your own whispers are
focused back on your head, so that they sound loud to you. Given your
distance to the surface, what is the surface's focal length?
Find the focal length of the mirror in problem 5 of chapter 1.
Rank the focal lengths of the mirrors, from shortest to longest.
(a) A converging mirror is being used to create a virtual image. What is
the range of possible magnifications? (b) Do the same for the other types
of images that can be formed by curved mirrors (both converging and
(a) A converging mirror with a focal length of 20 cm is used to create
an image, using an object at a distance of 10 cm. Is the image real, or is it
virtual? (b) How about f=20 cm and do=30 cm? (c) What if it was a
diverging mirror with f=20 cm and do=10 cm? (d) A diverging mirror with
=20 cm and do=30 cm?