Question

Astronomers are proud of their telescopes and often say they can "resolve a dime at so many miles." How many miles from a 102 cm telescope mirror can you place a coin of diameter 3.0 cm and just resolve the coin using light of wavelength 389 nm? (1 mi = 1.609 km) Astronomers consider an object "resolved" if the angular scale of the object's radius is equal to the resolving limit of the telescope.

Answer 1

To find the distance at which a coin can be resolved by a telescope, we can use the formula θ = 1.22 × (λ / D). By substituting the values into the formula and solving, we find that the coin can be resolved at a distance of approximately 0.0398 miles from the telescope mirror.

Step-by-step explanation:

To calculate the minimum distance from the telescope mirror to resolve an object, we can use the formula:

θ = 1.22 × (λ / D)

Where θ is the angular scale, λ is the wavelength of light, and D is the diameter of the telescope mirror.

1. Convert the diameter of the coin from cm to m: 3.0 cm ÷ 100 = 0.03 m
2. Convert the wavelength of light from nm to m: 389 nm ÷ 10^9 = 3.89 × 10^-7 m
3. Convert the diameter of the telescope mirror from cm to m: 102 cm ÷ 100 = 1.02 m
4. Substitute the values into the formula: θ = 1.22 × (3.89 × 10^-7 m / 1.02 m) = 4.67 × 10^-7 radians
5. Convert the radians to degrees: 4.67 × 10^-7 radians × (180 ÷ π) = 0.027 degrees
6. Now we can find the distance using the formula:

d = diameter of the object / tan(θ)

1. Substitute the values into the formula: d = 0.03 m / tan(0.027 degrees)
2. Calculate the tangent of the angle in degrees: tan(0.027 degrees) = 0.000469
3. Calculate the distance: d = 0.03 m / 0.000469 = 64.02 m
4. Convert the distance to miles: 64.02 m ÷ 1609 = 0.0398 miles

Therefore, you can place a coin of diameter 3.0 cm at a distance of approximately 0.0398 miles from a 102 cm telescope mirror and still resolve the coin using light of wavelength 389 nm.