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Suppose you are standing on a straight highway and watching a car moving away from you...

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medusaaaaaaaaa | Student | (Level 1) Honors

Posted March 27, 2013 at 3:53 AM via web

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Suppose you are standing on a straight highway and watching a car moving away from you at 20.0 m/s. The air is perfectly clear, and after 11 minutes you see only one taillight. If the diameter of your pupil is 7.00 mm and the index of refraction of your eye is 1.33, what is the distance between the taillights. Taillights are often red so you can assume that λ=700 nm. (Hint: Assume the eye is acting like a single slit with a circular aperture. You will also need to take into consideration the fact that the wavelength of light changes inside your eye since the index of refraction is different. Please show all steps.)

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valentin68 | College Teacher | (Level 3) Associate Educator

Posted October 14, 2013 at 3:49 PM (Answer #1)

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The angular separation resolution of a lens is given by

`theta =1.22*lambda/D` ,  (1)

where `D =7 mm` is the diameter of the lens.

The speed of light is smaller for higher refraction indexes than 1:

`c/v =n`

which means a shorter wavelength in a medium having a refraction index `n>1`

`lambda =v*T =c/n*T = lambda_0/n`

Combining this with (1) one gets

`theta = 1.22*lambda_0/(n*D)`

The distance from the person to the car is

`L =v*t =20*11*60 = 13200 m`

If `l` is the distance between the taillights one has

`l/L =tan(theta)~~theta =1.22*lambda_0/(nD)`

Thus the for the distance `l` one gets the value

`l =(1.22*lambda_0*L)/(n*D) = (1.22*700*10^-9*13200)/(1.33*0.007) = 1.21 m`

The distance between the taillights is 1.21 m

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