lilkrazyrae said:
Ok once again I'm lost you end up with r^2 canceling out too I cannot find anyother way to set it up
inha's hint is very useful. You assumed that the two measures of radius are equal when in fact they are not. Well...they are at d/2 (according to inha's variables).
Initially we told you that
F_{gravity}=\frac{Gm_1m_2}{r^2}
So it follows that:
F_{earth}=\frac{Gm_{ship}m_{earth}}{r^2}
where
r equals the distance from the Earth to the ship. And it also follows that:
F_{moon}=\frac{Gm_{ship}m_{moon}}{r^2}
but the
r in
this equation is the distance from the
moon to the ship.
You want to find out when these two equations equal each other, or in other words:
F_{moon}= F_{earth}
So begin by setting their equations equal to each other (I'm sure you've already done this, but pay attention to the variables...same letters but different subscripts):
\frac{Gm_{ship}m_{moon}}{r_{m}^2}=\frac{Gm_{ship}m_{earth}}{r_{e}^2}
where r_{m} is the distance between the ship and the moon, and r_{e} is the distance between the ship and the earth. Remember that the mass of the moon is only 0.0123 the mass of the earth. That expresses the mass of the moon
in terms of the mass of the earth, so we can eliminate the m_{moon} and express it in Earth masses. That means:
\frac{Gm_{ship}0.0123m_{earth}}{r_{m}^2}=\frac{Gm_{ship}m_{earth}}{r_{e}^2}
Notice how the variables start to drop off. G, the mass of the ship, and the mass of the Earth m cancel out now:
\frac{0.0123}{r_{m}^2}=\frac{1}{r_{e}^2}
Getting it? I hope I haven't done too much, lol! But I'm sure your confusion was because you weren't keeping in mind that, even though the variable letters all represent similar IDEAS, you should not assume that they represent the same QUANTITIES OF THOSE IDEAS in this problem. If you have two different distances you should use two different variables. In this case we use
r, but qualify it further with a subscript. Heck, you don't have to use subscripts if you use an entirely different letter, but the r's plug into the gravity equation nicely. It's fair enough to rewrite this equation a new way:
\frac{Gm_{ship}m_{moon}}{r_{m}^2}=\frac{Gm_{ship}m _{earth}}{r_{e}^2}
\frac{Gab}{d^2}=\frac{Gae}{f^2}
Where
a = mass of the ship
b = mass of the moon
d = distance between ship and moon
e = mass of the earth
f = distance between ship and earth
Notice that you don't run into the problem you had earlier, with the r^2 cancelling out here, because you can't cancel f's with d's!
