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.


020 (part 2 of 2) 10.0 points
What would the acceleration of the astronaut 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/s

Answer 1

Answer:

a) rM<3.456* 10e8 m

b) a= 3.33 · 10e-3 m/s²

Explanation:

mA -mass of the astronaut

mE=5.98*10 24 kg mass of the Earth

mM= 7.36* 10 22kg mass of the Moon

r -distance between Eart and Moon, r= 3.84*10 8 m

rM - the point where the forces from the Earth and the Moon are the same

g gravitanioal const5ant g=6.67*10 -11 Nm2/kg2

FE=FM

g*mE*mA/(r-rM)2=g*mM*mA/(rM2) yields

mE/(r-rM)2=mM/rM2

sqrt (mE/mM)*(r-rM)=rM

rM*(1+sqrt (mE/mM))=sqrt (mE/mM)*r

rM=(sqrt (mE/mM)*r)/(1+sqrt (mE/mM))

rM=9*3.84*10 8m/10

rM=3.456* 10 8 m

less than that distance is the asked one

b) F=Fg

mA*a=g*mA*mE/(rM2)

a=g*mE/(rM2)

a=6.67*10 -11 Nm2/kg2 *5.97*10 24kg/ (3.456 10 8m)2

a=3.33* 10 -3 m/s2