While flying at an altitude of 7.25 km, you look out the window at various objects on the ground. If your ability to distinguish two objects is limited only by diffraction, find the smallest separation between two objects on the ground that you can resolve as distinguishable. Assume your pupil has a diameter of 4.0 mm and take λ = 570 nm.
To find the smallest separation between two objects on the ground that you can resolve as distinguishable, we can use the formula for the angular resolution of a telescope:
θ = 1.22 * (λ / D)
where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the pupil.
First, let's convert the altitude from kilometers to meters:
Altitude = 7.25 km = 7250 m
Next, we can calculate the distance between the ground and the observer using basic trigonometry:
Distance = square root of (altitude^2 + earth's radius^2) - earth's radius
The radius of the earth is approximately 6,371 km = 6,371,000 m.
Distance = square root of (7250^2 + 6,371,000^2) - 6,371,000
≈ square root of (52,531,250,000 + 40,558,241,000,000) - 6,371,000
≈ square root of 40,610,492,250,000 - 6,371,000
≈ square root of 40,610,485,879,000
≈ 6,370,915.10 m
Now let's calculate the angular resolution using the formula:
θ = 1.22 * (λ / D)
Substituting the given values:
θ = 1.22 * (570 nm / 4.0 mm)
Note that 1 mm = 1,000,000 nm.
Converting the values:
θ = 1.22 * (570 / 4,000)
≈ 0.1746 radians
Finally, let's calculate the smallest separation between two objects on the ground:
Separation = Distance * θ
Separation = 6,370,915.10 * 0.1746
≈ 1,111,135.97 m
Therefore, the smallest separation between two objects on the ground that you can resolve as distinguishable is approximately 1,111,136 meters.
To determine the smallest separation between two objects on the ground that you can distinguish, we can use the concept of diffraction-limited resolution.
The formula for the minimum resolvable angle for two objects separated by a distance, d, is given by:
θ = 1.22 * (λ / D),
where:
θ is the angular resolution,
λ is the wavelength of light, and
D is the diameter of the pupil.
In this case, we can convert the separation distance between objects on the ground to an angle using the concept of similar triangles.
Given that the altitude (h) from which you are observing is 7.25 km and the separation (d) between the objects on the ground is the smallest resolvable separation, we can set up a right triangle:
tan(θ) = (d/2) / h.
Rearranging the equation, we get:
d = 2 * h * tan(θ).
Now, substituting the value of θ from the previous formula, we have:
d = 2 * h * tan(1.22 * (λ / D)).
Plugging in the values:
h = 7.25 km = 7250 m,
λ = 570 nm = 570 x 10^(-9) m, and
D = 4.0 mm = 4 x 10^(-3) m,
we can calculate the smallest separation between the two objects on the ground:
d = 2 * (7250 m) * tan(1.22 * (570 x 10^(-9) m) / (4 x 10^(-3) m)).
Using a calculator, the final value of d is approximately 2.044 meters.
Therefore, the smallest separation between two objects on the ground that you can resolve as distinguishable is approximately 2.044 meters.