Where Should the Certificate Be Awarded on a Space Tour from Earth to Moon?

[ML1]

Homework Statement

Transplanertay tours promises tour participants a certificate to commemorate their passage from the stronger influence of Earth's graviational pull to the stronger pull of the Moon at the point where the two forces on your spaceship are equal. Where on the trip should you award the certificate?

Homework Equations

F_G = G (m_1)(m_2)/ (r^2)

The Attempt at a Solution

We know that F will equal zero right...? I'm confused how to go about it. I know I'm solving for r but if F= 0 i can't solve for anything.

 

[collinsmark]

We know that F will equal zero right...?

Yes. [Edit: Well, at least the component of the resulting force that is parallel to the line intersecting the Earth and the Moon will be zero. If the ship is not on that line, there will be non-zero component of the net force pushing it toward that line.]

I'm confused how to go about it. I know I'm solving for r but if F= 0 i can't solve for anything.

There are two rs involved. There is the distance from the center of the Earth to the ship r_{earth_ship}, and the distance from the ship to the center of the Moon r_{ship_moon}.

If the ship happens to be on a line intersecting the Earth and Moon, These two distances are related by the distance from the center of the Earth to the center of the Moon d_{earth_moon}.

d_{earth_moon} = r_{earth_ship} + r_{ship_moon} *​

*(Again though, the above equation is a special case where the ship is on a straight line between the Earth and the Moon. If you keep things in terms of both r_{earth_ship} and r_{ship_moon}, you don't need to use the above equation, and your answer will apply more generally.)

In general, there are two forces on the ship we are concerned with. There is the force of gravity from the Earth and there is the force of gravity from the Moon. It is the sum of these equal and opposite force magnitudes that equals zero -- not just anyone particular magnitude.

[Edit: in other words, set the magnitude of the Earth's gravitational force on the ship equal to the Moon's gravitational force magnitude on the ship, and simplify.]

 

[ML1]

[Edit: in other words, set the magnitude of the Earth's gravitational force on the ship equal to the Moon's gravitational force magnitude on the ship, and simplify.]

Like this...?

F (M_e)(M_s) / (R_es)^2 = F (M_s)(M_m) / (R_ms)^2 ?



es - Earth/ship

ms - moon shop

 

[collinsmark]

Like this...?

F (M_e)(M_s) / (R_es)^2 = F (M_s)(M_m) / (R_ms)^2 ?



es - Earth/ship

ms - moon shop

Yes, except remove the "F" from both sides of the equation.

You can also simply further too. Notice that M_s cancels out.