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Energy required to go from one planet to another

  1. Apr 26, 2013 #1
    1. Description of Problem
    A binary planet system comprises two identical planets of mass M and radius R with their centers a distance 10 R apart. The minimum energy that the engines of a spacecraft need to supply to get a rocket of mass m from the surface of one planet to the surface of the other is of the form XGMm/R. What is X?

    2. Relevant equations
    Escape Velocity v= √2GM/R
    Gravitational Kinetic Energy KE=GMM/2R
    Gravitational Potential Energy U = GmM/R
    Kinetic Energy = 1/2mv2

    3. Attempt at a solution

    use escape velocity to find the KE of escape velocity is KE = mMG/R

    I'm not sure where to go from here...

    subtract the potential energy that the second planet supplies once the rocket reaches halfway?

    this gives x=.8
     
  2. jcsd
  3. Apr 26, 2013 #2

    mfb

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    I think you are supposed to make some approximations, otherwise the problem becomes really complicated.
    - neglect orbital mechanics and how rockets work
    - neglect the motion of the planets around each other?

    If you just consider the potential energy at the surface of a planet and at the center between the planets, you get some nice answer (it is not 0.8, however - don't forget the potential of the other planet).
     
  4. Apr 26, 2013 #3
    What we need to find is the amount of energy to get the rocket half way to the other planet, right? this can be found y finding the difference between the potential energies at the surface and half way through?

    Potential at the surface is MmG/R - MmG/9R = 8MmG/9R
    The second term is the Potential E from the second planet - I didn't include this in my previous attempt

    potential halfway is MmG/5R

    this gives X=.688

    This is close to the answer which is listed as X=0.7

    Is this correct?
     
    Last edited: Apr 26, 2013
  5. Apr 26, 2013 #4
    I understand this "Potential at the surface is MmG/R - MmG/9R = 8MmG/9R"

    How did you get this "potential halfway is MmG/5R"
     
  6. Apr 26, 2013 #5
    Halfway between the center of the two planets is 5R, so the potential E from the second planet at half way is MmG/5R

    Conceptually, the potential at that point is 0 because the planets pull on it equally?
     
  7. Apr 26, 2013 #6

    mfb

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    And the potential from the other planet?

    The force is 0, the potential is not (with your definition of the potential).
     
  8. Apr 26, 2013 #7
    the potential from the other planet is the same...

    so the potential at the midpoint is 2MmG/5R?
     
  9. Apr 27, 2013 #8

    mfb

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    I think there is a minus sign missing everywhere, but apart from that: right.
     
  10. Apr 27, 2013 #9
    How can there be a potential if there is no force?
    Also, consider one of your previous equations,.. Potential at the surface is MmG/R - MmG/9R = 8MmG/9R ,
    using this, the potential halfway between would be zero because the first term would change to MmG/5R as well as the second term and the difference would be zero it seems.
     
  11. Apr 27, 2013 #10

    mfb

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    There is nothing wrong with that. Actually, every potential minimum, maximum* and critical point has no (net) force, while every point in space has a potential.

    *does not exist in gravity
    There is a sign error, too.
     
  12. Apr 27, 2013 #11
    If you do the calculus and integrate the f dl from R to 5R you get 8MnG/9R, so I think this would be the correct answer. In a similar manner if you integrate from R to infinity you will get 0,8, not 8/9. I am a bit confused but will figure this out eventually, unless someone shows me the way.
     
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