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The perfect spot between the Earth and the Moon

  1. Mar 24, 2015 #1
    1. The problem statement, all variables and given/known data
    There exists a spaceship orbiting between the Moon and the Earth. Find the perfect spot where the gravity due to Earth is equal to the one caused by the Moon. I have already solved for the ratio ## \frac {R_E}{\sqrt{M_E}} = \frac {R_M}{\sqrt{M_M}} ##. ## R_E ## is the distance between the spaceship and the Earth while ##R_M## represents the distance between the ship and the Moon. Likewise ##M_M## and ##M_E## are masses. The next step in my solutions guide is ## R_E = \frac {(R_E + R_M)\sqrt{M_E}}{\sqrt{M_E}+\sqrt{M_M}} ##. I don't understand how they got to that.

    2. Relevant equations
    1) Correct answer ## = R_E = 3.46 * 10^8 ## meters
    2) Mass of the Moon ##=7.347*10^22##
    3) Mass of the Earth ##=5.98*10^24##
    4) Distance between Moon and Earth ##= R_E + R_M = 3.844*10^8##

    3. The attempt at a solution
    Using ratio I solved for in "The problem statement, all..." I tried to isolate ##R_E##
    ##R_E = \frac{R_M\sqrt{M_E}}{\sqrt{M_M}} ##
    From 4) from part 2 "Relevant equa..." I get that ##R_M = 3.844*10^8 - R_E##
    Therefore ##R_E = \frac{(3.844*10^8 - R_E)\sqrt{M_E}}{\sqrt{M_M}} ##
    Thus ##R_E = (3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}}) - R_E(\frac{\sqrt{M_E}}{\sqrt{M_M}}) ##
    ##R_E + R_E(\frac{\sqrt{M_E}}{\sqrt{M_M}}) = (3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}}) ##
    ##R_E(1+(\frac{\sqrt{M_E}}{\sqrt{M_M}})) = (3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}}) ##
    ##R_E = \frac{(3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}})}{1+(\frac{\sqrt{M_E}}{\sqrt{M_M}})}##
    While I did get the correct answer, my equation is so complicated. How do I turn it into the one given by the solutions manual?
     
  2. jcsd
  3. Mar 24, 2015 #2
    Latex refuses to load for some reason... Any solutions? Thanks everyone.
     

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    Last edited: Mar 24, 2015
  4. Mar 24, 2015 #3

    PeroK

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    Apart from using that specific number instead of ##R_E+R_M##, there is nothing different about your solution. All you have to do is multiply top and bottom by ##\sqrt{M_M}## to get the book answer.
     
  5. Mar 24, 2015 #4
    Now I see it! Thank you so much!
     
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