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 kilometers, are you marginally able to discern that there are two headlights rather than a single light source

Answer 1

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  $z = 4707.6 \ m$

Explanation:

From the question we are told that

    The is distance between the headlight is $d = 0.681 \ m$

   The wavelength is  $\lambda = 549 \ nm = 549 *10^{-9} \ m$

    The  pupil diameter is  $D = 4.63 \ mm = 0.00463 \ m$

Generally, we can mathematically evaluate the resolution of the eye as

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

    substituting values

              $\theta = \frac{1.22 * 549 *10^{-9} }{0.00463}$

              $\theta = (1.45 *10^{-4} )^o$

Now according to SOHCAHTOA rule  

         $sin \theta = \frac{ d}{z}$

Where z is  the distance at which the eye can discern the two head light

  given that the angle is very small $sin \theta = \theta$

=>    $\theta = \frac{ d}{z}$

substituting values

     $1.45*10^{-4} = \frac{ 0.681}{z}$

=>   $z = \frac{0.681}{1.45 *10^{-4}}$

=>    $z = 4707.6 \ m$