The writing on the passenger-side mirror of your car says \"Warning! Objects are

Question

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 -80 cm . A 1.5 m -tall cyclist on a bicycle is 20 m from the mirror. You are 1.4 m from the mirror, and suppose, for simplicity, that the mirror, you, and the cyclist all lie along a line.

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

What would the angular size of the cyclist's image have been if the mirror were flat?

Explanation / Answer

1/f = 1/di + 1/do
1/-0.80 = 1/di + 1/20

di = -0.769 m behind the mirror

so you are 1.4 + 0.769 = 2.169 m from the image

if flat mirror
the image would have been 20m behind so 1.4 + 20 = 21.4 m

hi/ho = di/do
hi/1.5 = 0.769 / 20

hi = 0.0576 m

same as object so 1.5 m

angle size = 2(arctan (1/2 hi/d) ) = 2(arctan(1/2*0.0576/2.169) = 1.52 degrees

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

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