Answer to Question #100567 in Physics for Nyx

Question

On the way to the moon, the Apollo astronauts reach a point where the Moon’s gravitational pull is stronger than that of Earth’s.
Find the distance of this point from the
center of the Earth. The masses of the
Earth and the Moon are 5.98 × 1024 kg and
7.36 × 1022 kg, respectively, and the distance
from the Earth to the Moon is 3.84 × 108 m.
Answer in units of m.

Answer 1

At the point where the Moon’s gravitational pull of the astronaut is stronger than that of Earth’s

$$F_{\rm EA}\le F_{\rm MA}$$

$$G\frac{M_E m_A}{x^2}\le G\frac{M_M m_A}{(L-x)^2}$$

$$\frac{5.98\times 10^{24}}{x^2}\le \frac{7.36\times 10^{22}}{(3.84\times 10^8-x)^2}$$

$$9\times(3.84\times 10^8-x)\le x$$

$$x\geq\frac{9\times 3.84\times 10^8}{10}=3.46\times 10^8\:\rm m$$