A car with its 2 headlights on is approaching you at night. Approximately how close (in meters) does the car have to be to you so that you can distinguish the two headlights? Assume that the diameter of your pupils is 6.5 mm, the distance between the two lights is 1.5 m, and the wavelength of the light is 550 nm.
Any one
To determine the distance at which you can distinguish the two headlights, we can consider the concept of angular resolution. Angular resolution is a measure of how well our eyes (or any optical instrument) can distinguish between two separate points or objects.
The formula to calculate angular resolution is given by:
Angular Resolution (in radians) = 1.22 * (wavelength of light) / (diameter of the pupil)
Let's plug in the values into the formula:
Wavelength of light (λ) = 550 nm = 550 * 10^(-9) m
Diameter of the pupil (D) = 6.5 mm = 6.5 * 10^(-3) m
Angular Resolution = 1.22 * (550 * 10^(-9) m) / (6.5 * 10^(-3) m)
Calculating this value gives us the angular resolution in radians. To convert it to meters, you can multiply it by the distance between your eyes and the object. However, we'll assume that the distance between your eyes and the car is much larger than the distance between the car's headlights, so we can approximate the distance as infinity.
Therefore, the angular resolution becomes the actual resolution in meters.
Now, the approximate distance at which you can distinguish the two headlights is:
Distance ≈ Diameter of the pupil / Angular Resolution
Plugging in the values:
Distance ≈ (6.5 * 10^(-3) m) / [1.22 * (550 * 10^(-9) m) / (6.5 * 10^(-3) m)]
Simplifying further:
Distance ≈ 1.5 m
So, the approximate distance at which you can distinguish the two headlights is approximately 1.5 meters.