Question

A spaceship of mass m travels from the Earth to the Moon along a line that passes through the center of the Earth and the center of the Moon.
(a) At what distance from the center of the Earth is the force due to the Earth three times the magnitude of the force due to the Moon?

Answer 1

You will need the ratio of the masses of earth and moon, Me and Mm, and the distance between earth and moon, D. The spacecraft mass m will not matter.

Let the earth-to-spaceship distance be x.

Me/x^2 = Mm/(D-x)^2

Mm/Me = [(D-x)/x]^2 = [(D/x) -1]^2

Solve for D/x, and from that, get x.