(1) A virtual image -7 cm tall and 17 cm away from a mirror is created by an object 7 cm away from a mirror. How tall is the object? What kind of mirror created the image? What is its focal length?
(2) If a 1.75 m tall man is looking at a plane mirror from an eye which is 15 cm from the top of their head, what is the minimum length that the mirror must be so that he can just barely see his whole body in the mirror? How far up from the ground must the mirror be placed?
(1) |(object size)/(image size)|
= |(object distance)/(image distance)|
(use absolute values above)
object size/7 = 7/17
object size = 49/17 = 2 15/17
1/f = -1/17 + 1/7
f = 17*7/(7-17) = -11.9 cm
A convex mirror formed the image.
The minus sign is in front of 1/17 because it is a virtual image
Draw an image diagram with three vertical lines spaced equal distances apart. (The separation distance does not matter, as long as they are equal). The three vertical lines represent the man, the mirror and the virtual image. The man and image will have equal height, at the left and the right. Mark a spot near the top of the left-most line representing his eye, 1.60 m up from the bottom.
A ray from his toes that gets reflected reaches his eye will strike the mirror 0.8 m from the floor. A ray from the top of his head that gets reflected will strike the mirror 1.675 m from the floor. There must be mirror between these to points to be able to see the entire body length. Drawing the figure will show you why and what the two answers are.
To find the answers to these questions, we can use the mirror equation and magnification formula. The mirror equation relates the object distance (o), the image distance (i), and the focal length (f) of the mirror:
1/(f) = 1/(o) + 1/(i)
The magnification formula relates the object height (h_o), the image height (h_i), and the magnification (m):
m = -(h_i) / (h_o)
Now let's use these formulas to solve the questions:
(1) A virtual image -7 cm tall and 17 cm away from a mirror is created by an object 7 cm away from a mirror.
To determine the height of the object, we need to find the magnification first:
m = -h_i / h_o
-7 cm / h_o = m
Next, using the mirror equation, we can find the image distance (i):
1/f = 1/o + 1/i
1/f = 1/7 cm + 1/i
Since it is mentioned that the image is virtual, we know that the mirror must be a concave mirror.
Let's substitute the values into the equation:
1/f = 1/7 cm + 1/i
1/f = (i + 7) / (7i)
7i / f = i + 7
7i = fi + 7f
7i - fi = 7f
i(7 - f) = 7f
i = 7f / (7 - f)
Now, we can substitute this value of i into the magnification formula:
-7 cm / h_o = 7f / (7 - f)
Solving this equation for h_o, we can find the height of the object.
To find the value of the focal length (f), we can use the information given in the problem, where the object distance (o) is 7 cm and the image distance (i) is 17 cm.
(2) If a 1.75 m tall man is looking at a plane mirror from an eye which is 15 cm from the top of their head, we need to determine the minimum length that the mirror must be so that he can just barely see his whole body in the mirror. We also need to find how far up from the ground the mirror must be placed.
To determine the minimum length of the mirror, we need to calculate the maximum height of the image. Since the man wants to see his whole body, the height of the image (h_i) should be equal to the height of the man (1.75 m).
Using the mirror equation:
1/f = 1/o + 1/i
Assuming that the mirror is placed vertically, the object distance (o) would be the distance from the man's eye to the top of his head minus the height of the man (15 cm - 1.75 m).
We can now solve this equation to find the image distance (i), which will help us determine the minimum length of the mirror.
To find the height above the ground the mirror must be placed, we need to consider the eye height of the man. We can calculate this by subtracting the height of the man from the distance between the ground and the top of his head.
Now we have the necessary information to solve the problem.