People who are really really good at Physics...please help me!?

Question

On the way to the moon the Apollo astronauts reach point where the Moon’s gravitational pull is stronger than that of Earth’s.
Determine the distance of this point from the center of the Earth. The masses of the Earth and the Moon are respectively 6.07×10^24 kg and 7.36x 10^22 kg. The distance from the Earth to the Moon is 3.6x10^8 m. Correct answer: 3.24291x 10^8 m.

Our teacher gave us the answer, and we are supposed to show the work. I don't get it though. I started off by writing the equation R=G x m x Mass of Earth all over r^2 + G x m x Mass of moon all over r^2, but my teacher said this was wrong. I really don't get it. She said that the things that I put =R have to be equal to each other instead.

Does anyone understand this?

Answer 1

F = (G)(Ma)(Mb) / r^2

Let Mb be the mass of the Apollo space craft, astronauts, etc
Let x be the distance from Earth to the space craft
Thus the distance from the Moon to the space craft is 3.6x10^8 - x

Force of gravity from Earth
F = (G)(Mearth)(Mb)/x^2

Force of gravity from t3.6he Moon
F = (G)(Mmoon)(Mb)/ (3.6x10^8-x)^2

Since you are looking for the point where the forces are equal
(G)(Mearth)(Mb)/x^2 = (G)(Mmoon)(Mb)/ (3.6x10^8-x)^2
and
(Mearth)/x^2 = (Mmoon)/ (3.6x10^8-x)^2
thus
(Mearth) (3.6x10^8-x)^2) = (Mmoon) (x^2)

and the rest is just calculation

Answer 2

Yes, I understand.

You need to find the distance R, where the gravity pull of the Moon and the gravity pull of the Earth are equal in value, but opposite in direction.

Gravity from Moon = Gravity from Earth
G m / (EM distance - R)^2 = G M / R^2

Divide both sides by G, you get:
m / (EM distance - R)^2 = M / R^2

You have numbers for everything except R, so you can solve for that.

Answer 3

where you put R, that is the FORCE.

Force of earth on dudes = G m Mass earth / distance to earth squared.

Force of moon on dudes = G m Mass moon / distance to moon squared.

You are looking for the point where those two are equal. So set distance to moon = distance of the whole trip minus distance to earth. Then solve.

Answer 4

mm = mass of moon, me = mass of earth, rm = distance of object from moon, re = distance of object from earth (centre)

then force applied by earth on object = G x m x me / re^2
force applied by moon on object = G x m x mm / rm^2

Since these forces are given to be equal, this is what your teacher is saying, put these formulae as equal.

Then cancel out G & m, and rerrange the formulae should give you :

re^2 = rm^2 * me/mm. re = 9.08 rm

since rm+re = 3.6 x 10^8, re = 3.24 * 10^ 8 metres and re = 3.6 *10 ^ 6 metres