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 twice the magnitude of the force due to the Moon? (b) How does your answer to part (a) depend on the mass of the spaceship? Explain

Answer 1

We have to find the distance at which the force due to earth is twice the force due to moon.
Let x be the distance between the earth and the spaceship at which the force due to earth is twice the force due to moon.
The distance between the spaceship and the moon at that moment is (r -x), where r is the total distance between Earth and Moon.
F_e = F_m
G M_e m / x² = G M_m m / (r -x)² ------------------------ 1
Where m is the mass of the spaceship.
M_e / M_m = (r - x)² / x²
5.972 × 10²⁴ kg / 7.34767309 × 10²² kg = (384,400 x 10³ - x )² / x2
(384,400 x 10³ - x ) / x = 9.0154
384,400 x 10³ =  10.0154 x
x =  38380.887 km
b) Answer to part a does not depend on mass of the spaceship. In equation 1 it can be seen that mass of the spaceship cancels out from both side of the equation.