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 7.0 mm. The taillights of this car are separated by a distance of 1.2 m and emit red light (wavelength = 669 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?

Well, let's see here! If the car's taillights start to merge into a single spot of light due to diffraction, it means that the angle formed by the light entering your eye needs to be larger than the angular resolution of your eye. And since you have some mighty pupils of 7.0 mm, we can estimate that your eye's angular resolution is about 1 degree.

To find the distance to the car when the taillights merge, we can use a bit of trigonometry. The formula is:

Distance = (Separation between taillights) / (2 * tangents(Angular resolution))

Given that the separation between the taillights is 1.2 m and the angular resolution is 1 degree, we can plug in these values to find our answer:

Distance = 1.2 m / (2 * tangent(1 degree))

Now, I can crunch the numbers for you, but please remember that this is just an estimate using some simplifications. According to my calculations, the car would be roughly 125 meters away from you when its taillights appear to merge into a single spot of light due to diffraction. So, keep your eyes open, but more importantly, keep your sense of humor intact while on the road! Safe travels!

To determine the distance at which the taillights of the car appear to merge into a single spot of light due to diffraction, we can use the formula for the angular resolution of the human eye:

θ = 1.22 * λ / D

Where:
- θ is the angular resolution in radians
- λ is the wavelength of light in meters
- D is the diameter of the pupils of the eye in meters

Given:
- Pupil diameter (D) = 7.0 mm = 0.007 m
- Wavelength (λ) = 669 nm = 669 x 10^-9 m

Substituting the given values into the formula, we can calculate the angular resolution of the eye:

θ = 1.22 * (669 x 10^-9) / (0.007)

θ ≈ 0.000114 radians

Now, let's consider a right triangle formed by your eyes, the car, and the point where the taillights appear merged. The angle formed by your eyes, the merged taillights, and the car is equal to θ.

We can use trigonometry to find the distance (d) at which the taillights appear merged:

tan(θ) = (1.2 / 2) / d

Rearranging the equation, we get:

d = (1.2 / 2) / tan(θ)

Substituting the calculated value of θ, we have:

d ≈ (1.2 / 2) / tan(0.000114)

d ≈ 1.046 x 10^4 meters

Therefore, the car is approximately 10,462 meters away when its taillights appear to merge into a single spot of light due to diffraction.

To determine the distance at which the car's taillights appear to merge into a single spot of light due to diffraction, we can use the concept of the diffraction limit. The diffraction limit is given by the formula:

θ = 1.22 * λ / D

where:
θ = angular size of the diffraction limit
λ = wavelength of light
D = diameter of the pupil of your eyes

First, let's convert the diameter of your pupils to meters:
D = 7.0 mm = 7.0 x 10^-3 m

Now, let's substitute the given values into the formula:
θ = 1.22 * 669 nm / 7.0 x 10^-3 m

Next, we need to convert the wavelength of light from nanometers to meters:
λ = 669 nm = 669 x 10^-9 m

Now we can calculate the angular size:
θ = 1.22 * (669 x 10^-9 m) / (7.0 x 10^-3 m)

Simplifying the expression:
θ = 8.949 x 10^-7 radians

We can now use this angular size to determine the distance at which the taillights merge into a single spot of light. The angular size is given by the formula:

θ = opposite / adjacent

where:
opposite = distance between the taillights
adjacent = distance from you to the car

We know the distance between the taillights is 1.2 m. Let's substitute this into the formula and solve for adjacent:

8.949 x 10^-7 radians = 1.2 m / adjacent

Rearranging the equation to solve for adjacent:
adjacent = 1.2 m / (8.949 x 10^-7 radians)

Calculating the value:
adjacent = 1.34 x 10^6 m

Therefore, the car is approximately 1.34 million meters away from you when its taillights appear to merge into a single spot of light due to diffraction.

angular spot size of each tail light =

(lambda)/d = 669*10^-9/7*10^-3 = 9.6*10^-5 radians

This equals the tail light spot angular separation at distance X, if
(1.2 m)/X = 9.6*10^-5
X = 13000 m = 13 km