You are looking into the headlights of a distant vehicle on an autoroute in France. The driver has shown his resistance to the Europeanization of Europe by resolutely continuing to use yellow headlights (wavelength 550. nm).
a) What is the theoretical limit of angular resolution of your eyes for these headlights when your iris has a diameter of 2.00 mm?
b) If the headlights of the car are separated by 1.80 m, with this resolution at what distance would you be just be able to tell that the vehicle had, in fact, two headlights?
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To calculate the theoretical limit of angular resolution of your eyes, we can use the formula:
θ = λ / D
where:
θ is the angular resolution,
λ is the wavelength of light, and
D is the diameter of your iris.
a) Let's calculate the angular resolution of your eyes:
Given:
λ = 550 nm = 550 x 10^-9 m
D = 2.00 mm = 2.00 x 10^-3 m
Substituting the values into the formula, we have:
θ = (550 x 10^-9 m) / (2.00 x 10^-3 m)
≈ 2.75 x 10^-4 radians
The theoretical limit of angular resolution of your eyes for these headlights is approximately 2.75 x 10^-4 radians.
b) To determine the distance at which you can discern two headlights, we can use trigonometry and the angular resolution.
Let D be the distance between the headlights.
The angle subtended by the headlights can be approximated by:
α = D / d
where:
α is the angle subtended by the headlights, and
d is the distance from your eyes to the vehicle.
We want to find the distance d at which α equals the angular resolution (θ), so we have:
θ = α
Substituting the values, we can rearrange the equation to solve for d:
d = D / θ
Given:
D = 1.80 m
θ ≈ 2.75 x 10^-4 radians
Substituting the values, we have:
d = (1.80 m) / (2.75 x 10^-4 radians)
≈ 6545.45 m
Therefore, at a distance of approximately 6545.45 meters, you would just be able to tell that the vehicle has two headlights.