Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 42 | Concept Questions A small mirror is attached to a vertical wall, and it hangs a distance y above the floor. A ray of sunlight strikes the mirror, and the reflected ray forms a spot on the floor. (a) From a knowledge of y and the horizontal distance x from the base of the wall to the spot, describe how one can determine the angle of incidence of the ray striking the mirror. If it is morning and the mirror is facing due east, would (b) the angle of incidence and (c) the distance x increase or decrease in time? Why?
Problem Suppose the mirror is 1.80 m above the floor. The reflected ray of sunlight strikes the floor at a distance of 3.86 m from the base of the wall. Later in the morning, the ray is observed to strike the floor at a distance of 1.26 m from the wall. The earth rotates at a rate of 15.0° per hour. How much time (in hours) has elapsed between the two observations?

| 43 | Concept Questions (a) Suppose that you are walking perpendicularly toward a stationary plane mirror. Following the method discussed in Section 3.4, express your image’s velocity relative to you in terms of the image’s velocity relative to the mirror and the mirror’s velocity relative to you. (b) How is the mirror’s velocity relative to you related to your velocity relative to the mirror? Explain. (c) Consider both velocities and . Do they have the same magnitudes and the same directions? Explain.
Problem When you walk perpendicularly with a velocity of +0.90 m/s toward a stationary plane mirror, what is the velocity of your image relative to you? The direction in which you walk is the positive direction.

| 44 | Concept Questions (a) For an inverted image that is in front of a mirror, is the image distance positive or negative and is the image height positive or negative? Explain. (b) Given the image distance, what additional information is needed to determine the focal length? Explain. (c) Given the object and image heights and a statement as to whether the image is upright or inverted, what additional information is needed to determine the object distance?
Problem A small statue has a height of 3.5 cm and is placed in front of a concave mirror. The image of the statue is inverted, 1.5 cm tall, and is located 13 cm in front of the mirror. Find the focal length of the mirror.

| 45 | Concept Questions These questions refer to Figure 25-22a. (a) As the object distance increases, does reflected ray 1 change? (b) As the object distance increases, does reflected ray 3 make a greater or smaller angle with respect to the principal axis? (c) Extending the reflected rays 1 and 3 behind the mirror allows us to locate the top of the image. As the object distance increases, does the image height increase or decrease?
Problem A convex mirror has a focal length of − 27.0 cm. Find the magnification produced by the mirror when the object distance is 9.0 cm and 18.0 cm. Verify that your answers are consistent with your answers to the Concept Questions.

| 46 | Concept Questions (a) Suppose that you are walking toward a stationary plane mirror as in the drawing. The view is from above. Following the method discussed in Section 3.4, express your image’s velocity relative to you in terms of the image’s velocity relative to the mirror and the mirror’s velocity relative to you. (b) How is the mirror’s velocity relative to you related to your velocity relative to the mirror? Explain. (c) Consider both velocities and . Do they have the same x and y components? Explain.

Problem You walk at an angle of toward a plane mirror, as in the drawing. Your walking velocity has a magnitude of 0.90 m/s. What is the velocity of your image relative to you (magnitude and direction)?

| 47 | Concept Questions A tall tree is growing across a river from you. You would like to know the distance between yourself and the tree, as well as its height, but are unable to make the measurements directly. However, by using a mirror to form an image of the tree, and then measuring the image distance and the image height, you can calculate the distance to the tree, as well as its height. (a) What kind of mirror, concave or convex, must you use? Why? (b) You will need to know the focal length of the mirror. The sun is shining. You aim the mirror at the sun and form an image of it. How is the image distance of the sun related to the focal length of the mirror? (c) Having measured the image distance di and the image height hi of the tree, as well as the image distance of the sun, describe how you would use these numbers to determine the distance and height of the tree.
Problem A mirror produces an image of the sun, and the image is located 0.9000 m from the mirror. The same mirror is then used to produce an image of the tree. The image of the tree is 0.9100 m from the mirror. (a) How far away is the tree? (b) The image height of the tree has a magnitude of 0.12 m. How tall is the tree?

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