Final answer:
Using the Rayleigh criterion for resolution, we find that the human eye can marginally distinguish two separate headlights that are 0.641 m apart at a distance of approximately 468 meters, which is 0.468 kilometers.
Step-by-step explanation:
To determine the distance at which a person can distinguish two separate headlights, we use the Rayleigh criterion for resolution, which states that the minimum angular resolution, θ, that the eye can discern is given by:
θ = 1.22 λ / D
where λ is the wavelength of light and D is the diameter of the pupil. The distance, d, at which the two points of light (the headlights) become resolvable is then calculated using the small angle approximation:
d = l / θ
where l is the distance between the two headlights.
Given λ = 549 nm = 549 × 10⁻⁹ m, D = 4.89 mm = 4.89 × 10⁻3 m, and l = 0.641 m, we can now solve for θ and consequently for d:
θ = 1.22 × 549 × 10⁻⁹ m / 4.89 × 10⁻3 m = 1.37 × 10⁻B radians
d = 0.641 m / (1.37 × 10⁻B radians) = 468 meters
Therefore, the distance in kilometers is 0.468 km, which is when you would be marginally able to discern that there are two separate headlights.