Question

You stand on a straight desert road at night and observe a vehicle approaching. This vehicle is equipped with two small headlights that are 0.657 m apart. At what distance, in kilometers, are you marginally able to discern that there are two headlights rather than a single light source? Take the wavelength of the light to be 549 nm and your pupil diameter as 4.63 mm.

Answer 1

Answer:

4.54168 km

Explanation:

Angular resolution is given by

$\theta=1.22\frac{\lambda}{D}$

where,

$\lambda$ = Wavelength = 549 nm

D = Diameter of pupil = 4.63 mm

$\theta=1.22\frac{549\times 10^{-9}}{4.63\times 10^{-3}}\\\Rightarrow \theta=0.00014466$

$sin\theta=\frac{d}{L}$ but for small angle $sin\theta=\theta$

where

d = Distance between the lights = 0.657 m

$\theta=\frac{d}{L}\\\Rightarrow 0.00014466=\frac{0.657}{L}\\\Rightarrow L=\frac{0.657}{0.00014466}\\\Rightarrow L=4541.68\ m$

The distance at which you will be able to marginally discern that there are two headlights rather than a single light source is 4.54168 km