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`

Step-by-step 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 = (1.22 * \lambda )/(D)`

substituting values

`\theta = (1.22 * 549 *10^(-9) )/(0.00463)`

`\theta = (1.45 *10^(-4) )^o`

Now according to SOHCAHTOA rule

`sin \theta = ( 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 = ( d)/(z)`

substituting values

`1.45*10^(-4) = ( 0.681)/(z)`

=>
`z = (0.681)/(1.45 *10^(-4))`

=>
`z = 4707.6 \ m`