When a spacecraft travels from the Earth to the Moon, both the Earth and the Moon exert a gravitational force on the spacecraft. Eventually, the spacecraft reaches a point where the Moon's gravitational attraction overcomes the Earth's gravity. How far from the Earth's surface must the spacecraft be for the gravitational forces from the Moon and the Earth to just cancel?

Question

When a spacecraft travels from the Earth to the Moon, both the Earth and the Moon exert a gravitational force on the spacecraft. Eventually, the spacecraft reaches a point where the Moon's gravitational attraction overcomes the Earth's gravity. How far from the Earth's surface must the spacecraft be for the gravitational forces from the Moon and the Earth to just cancel?

Answer 1

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Answer 2

To determine the specific distance from the Earth's surface where the gravitational forces from the Moon and the Earth cancel out, we need to consider the concept of gravitational force and understand the relationship between distance and gravitational force.

The gravitational force between two objects depends on two factors: the masses of the objects involved and the distance between them. The force of gravity is given by the equation:

F = G * ((m1 * m2) / r^2)

Where:
- F is the gravitational force
- G is the gravitational constant
- m1 and m2 are the masses of the two objects (in this case, the Earth and the Moon)
- r is the distance between the centers of the two objects

At the point where the gravitational forces from the Earth and the Moon cancel out, the magnitudes of the forces will be equal but act in opposite directions. Therefore, the equation becomes:

F(Earth) = F(Moon)

G * (m1 * m(spacecraft) / r^2(Earth)) = G * (m2 * m(spacecraft) / r^2(Moon))

Where m(spacecraft) represents the mass of the spacecraft.

Since the gravitational constant (G) and the mass of the spacecraft (m(spacecraft)) are the same on both sides of the equation, they cancel out. This simplifies the equation to:

m1 / r^2(Earth) = m2 / r^2(Moon)

We know that the mass of the Earth (m1) is greater than the mass of the Moon (m2). Therefore, to find the distance (r) from the Earth's surface where the gravitational forces from the Moon and the Earth cancel out, we can set up the following proportion:

r^2(Earth) / r^2(Moon) = m2 / m1

Now, substituting the known values:

(6400 km)^2 / r^2(Moon) = 1 / 0.0123

Simplifying this equation will allow us to solve for r^2(Moon), which will provide us with the distance from the Earth's surface to the spacecraft at the point where the gravitational forces cancel out.