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 0.673 m 0.673 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 543 nm 543 nm and your pupil diameter to be 5.37 mm.

Answer 1

To solve this problem we will apply trigonometric and optical concepts that allow us to obtain the minimum distance required. The resolution of the eye is given under the following condition,

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

Here,

$\lambda = Wavelength$

$D = Diameter$

With the values we have that the diameter will be,

$\theta = \frac{1.22(534nm)}{5.37mm}$

$\theta = 1.213*10^{-4}$

The relation between the distance of the lights and the distance from the eye to the lamp is given under the function,

$sin\theta = \frac{d}{L}$

For small angles $sin\theta = \theta$, then

$\theta = \frac{d}{L}$

Here,

d = Distance between lights

L = Distance from eye to lamp

$1.213*10^{-4} = \frac{0.673m}{L}$

$L = \frac{0.673m}{1.213*10^{-4}}$

$L = 5548.22m$

Therefore the distance will be 5.5km