You unconsciously estimate the distance to an object from the angle it subtends

Question

You unconsciously estimate the distance to an object from the angle it subtends in your field of view. This angle in radians is related to the linear height of the object hand to the distance d by = h/d. Assume that you are driving a car and that another car, 1.50 m high, is 24.0 m behind you. PLEASE provide thorough process thank you!!!

(a) Suppose your car has a flat passenger-side rearview mirror, 1.55 m from your eyes. How far from your eyes is the image of the car following you? (in meters)

(b) What angle does the image subtend in your field of view? (in rad)

(c) Suppose instead your car has a convex rearview mirror (see figures) with a radius of curvature of magnitude 2.20 m. How far from your eyes is the image of the car following you? (in meters)

(d) What angle does the image subtend at your eyes? (in rad)

(e) Based on its angular size, how far away does the following car appear to be? (in meters)

Explanation / Answer

from the mirror equation

1/P + 1/q = 1/f

1/P + 1/q = 2/R

1/24 + 1/q = 1/ inifinity

q = -24 m

distant from our eye is

1.55 m + 24 = 25.55 m

(b)

the image is te same size as the object

theta = h/d = 1.5/25.55 = 0.0587 rad

(c)

1/P+1/q = 2/R

1/24 + 1/q = 2/-2.2

q = -1.15 m

image distance is

1.55 m + 1.15 m = 2.70 m

(d)

m = hi/ho= -q/p

hi = -ho q/p = -1.5 ( -1.15 m/24 m) = 0.071875 m

theta' = hi/d = 0.071875/2.70 = 0.0266

(e)

d' = h/ theta' = 1.5 m/0.0266 =56.34 m

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