Final answer:
The point where the gravitational forces of the Earth and the Moon on a spacecraft cancel is closer to the Moon, and is found by setting the gravitational forces from both bodies equal and solving for the distance from the Earth to the spacecraft.
Step-by-step explanation:
The position where the gravitational forces exerted by the Earth and the Moon on a spacecraft cancel out is known as the Lagrange point or L1. This position is not halfway between the two bodies but is closer to the Moon due to the Moon's smaller mass relative to the Earth.
Mathematically, to find this point, we set the gravitational force exerted by the Earth Fe equal to the gravitational force exerted by the Moon Fm. The equation is as follows:
Fe = G(Mem/re2) = Fm = G(Mmm/dm2),
Where:
- G is the gravitational constant,
- Me and Mm are the masses of the Earth and the Moon,
- m is the mass of the spacecraft,
- re is the distance from the spacecraft to the center of the Earth,
- dm is the distance from the spacecraft to the center of the Moon,
- Note that re + dm is equal to the total distance between the Earth and the Moon.
By solving this equation, one could determine the point at which these forces balance, which is not at an equal distance because the Earth is much more massive than the Moon.