Final answer:
To find the distance between Uranus and its moon Miranda, we use Newton's Law of Universal Gravitation and the given gravitational force, masses of the two celestial bodies, and the gravitational constant to calculate the distance.
Step-by-step explanation:
To determine how far apart Uranus and its moon Miranda are, given the gravitational force between them, we can use Newton's Law of Universal Gravitation: F = G (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant (6.674 x 10^-11 N(m^2)/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Given:
- F = 2.28 x 10^19 N (force between Uranus and Miranda)
- m1 = 8.68 x 10^25 kg (mass of Uranus)
- m2 = 6.59 x 10^19 kg (mass of Miranda)
We can rearrange the equation to solve for r:
r = sqrt(G (m1 * m2) / F)
Rearranging and plugging in the known values:
r = sqrt((6.674 x 10^-11 N(m^2)/kg^2 * 8.68 x 10^25 kg * 6.59 x 10^19 kg) / 2.28 x 10^19 N)
After calculating this expression, we find the distance r between Uranus and Miranda.