Late one night on a highway, a car speeds by you and fades into the distance. Under these conditions the pupils of your eyes have diameters of about 7.3 mm. The taillights of this car are separated by a distance of 1.2 m and emit red light (wavelength = 676 nm in vacuum). How far away from you is this car when its taillights appear to merge into a single spot of light because of the effects of diffraction?

Distance between car and observer = 10619.5 m

Explanation:

First of all, we have to calculate the minimum angle which is given by the formula;

θ_min = 1.22λ/D

Where;

λ is wavelength = 676 nm = 676 x 10^ (9) m

D is diameter = 7.3 mm = 7.3 x 10^ (-3) m

Thus;

θ_min = (1.22 * 676 x 10^ (-9)) / (7.3 x 10^ (-3))

θ_min = 1.13 x 10^ (-4) rad

The distance between the object and the telescope has a relationship between the minimum angle and the separation distance and it's given by;

θ_min = d/L

Where d is separation distance = 1.2m and L is the distance between the object and the telescope

Making L the subject, we have;

L = d/θ_min

Thus;

L = 1.2 / (1.13 * 10^ (-4))

L = 10619.5 m