The writing on the passenger-side mirror of your car says “Warning! Objects 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 -80cm . A 1.5m -tall cyclist on a bicycle is 30m from the mirror. You are 1.4m 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?

e. What is the angular size of the image of the cyclist?

f. Why is there a warning label on the passenger-side mirror?

How am I supposed to go about solving this? Any help would be greatly appreciated. Thank you in advance.

Answer:

a

1/f = 1/di + 1/do

1/-0.80 = 1/di + 1/30

di = -0.779 m behind the mirror

so you are 1.4 + 0.779 = 2.18 m from the image

b

if flat mirror

the image would have been 30m behind so 1.4 + 30 = 31.4m

c

hi/ho = di/do

hi/1.5 = 0.779 / 30

hi = 0.03895 m

d

same as object so 1.5 m

e

angle size = 2(arctan (1/2 hi/d) ) = 2(arctan(0.3895/2.18) ) = 20.26°

f

since our brains are not set up to compensate for convex mirrors

we would percieve the cyclist to be much farther away than he actually is

because the image is much smaller

to our eyes it’s like we are looking in a flat mirror

the practical advantage to convex ‘spot’ mirrors

is the increased field of view