Consider a spaceship located on the Earth-Moon center line (i.e. a line that intersects the centers of both bodies) such that, at that point, the tugs on the spaceship from each celestial body exactly cancel, leaving the craft literally weightless. Take the distance between the centers of the Earth and Moon to be 3.72E+5 km and the Moon-to-Earth mass ratio to be 1.200E-2. What is the spaceship's distance from the center of the Moon?
Let R be the distance from Earth to Moon.
Let r be the distance from Earth to the spaceship, and R-r the distance from Moon to spaceship.
Then the force of gravity from spacecraft to Earth is EQUAL to the force of gravity from Spacecraft to Moon.
Set them equal, and solve for r.
To find the spaceship's distance from the center of the Moon, we can use the concept of gravitational forces between two bodies.
Let's denote:
- r as the distance from the spaceship to the center of the Moon,
- R as the distance from the spaceship to the center of the Earth,
- M as the mass of the Earth,
- m as the mass of the Moon,
- G as the gravitational constant.
According to the problem, the tugs on the spaceship from each celestial body exactly cancel, meaning that the gravitational forces on the spaceship due to the Earth and Moon are equal in magnitude but opposite in direction.
The gravitational force between two bodies is given by the formula:
F = G * (M * m) / r²
Since the forces from the Earth and Moon cancel out, we can set up the equation:
G * (M * m) / R² = G * (M * m) / r²
We can cancel out the gravitational constant (G), the mass of the Earth (M), and the mass of the Moon (m) from both sides of the equation:
1 / R² = 1 / r²
To solve for r, we can take the reciprocal of both sides:
R² = r²
Taking the square root of both sides, we get:
R = r
Therefore, the spaceship's distance from the center of the Moon is equal to its distance from the center of the Earth, which is 3.72E+5 km.