Question

On the way to the moon, the Apollo astro-
nauts reach a point where the Moon’s gravi-
tational 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

You must find the point of equilibrium between the two forces,
\[G\frac{M_Tm_s}{(R-x)^2}=G\frac{M_Lm_s}{x^2}\]
\[\frac{M_T}{(R-x)^2}=\frac{M_L}{x^2}\]So,
\[x=R\frac{\sqrt{M_LM_T}-M_L}{M_T-M_L}\]And R is the distance between Earth and Moon.

Answer 2

The result should be,
\[x=3.83\cdot10^7\ m\]from the center of the Moon, and
\[R-x=3.46\cdot10^8\ m\] from the center of the Earth.

Answer 3

What would the acceleration of the astro-
naut be due to the Earth’s gravity at this
point if the moon was not there? The
value of the universal gravitational constant
is 6.672 × 10−11 N · m2/kg2.
Answer in units of m/s2

Answer 4

As the distance from the center of the Earth is the number we found before,
\[d=R-x=3.46\cdot10^8\ m\]The acceleration at this point is,
\[g=G\frac{M_T}{d^2}=3.33\cdot10^{-3}\ m/s^2\]

Answer 5

Thank you so much :)

Answer 6

You're welcome.