Complete 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 0.681 m apart. At what distance, in kilometres, 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:
The distance is [tex]z = 4707.6 \ m[/tex]
Explanation:
From the question we are told that
The is distance between the headlight is [tex]d = 0.681 \ m[/tex]
The wavelength is [tex]\lambda = 549 \ nm = 549 *10^{-9} \ m[/tex]
The pupil diameter is [tex]D = 4.63 \ mm = 0.00463 \ m[/tex]
Generally, we can mathematically evaluate the resolution of the eye as
[tex]\theta = \frac{1.22 * \lambda }{D}[/tex]
substituting values
[tex]\theta = \frac{1.22 * 549 *10^{-9} }{0.00463}[/tex]
[tex]\theta = (1.45 *10^{-4} )^o[/tex]
Now according to SOHCAHTOA rule
[tex]sin \theta = \frac{ d}{z}[/tex]
Where z is the distance at which the eye can discern the two head light
given that the angle is very small [tex]sin \theta = \theta[/tex]
=> [tex]\theta = \frac{ d}{z}[/tex]
substituting values
[tex]1.45*10^{-4} = \frac{ 0.681}{z}[/tex]
=> [tex]z = \frac{0.681}{1.45 *10^{-4}}[/tex]
=> [tex]z = 4707.6 \ m[/tex]
