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Deriving Mechanics Equations for the Moon

  1. Jan 27, 2010 #1
    Alright, so I have a few questions.
    a) Consider a moon of mass m orbiting a planet of mass M in an elliptical orbit. The equation for the radius of an elliptical orbit is r=a(1-e2) / (1+ecos[tex]\vartheta[/tex]). Derive equations for the orbital radius of the moon when it closest to and farthest from the planet.
    My answer turns out to be periapse: r=a(1-e)
    and for apoapse: r=a(1+e) a is the semi major axis of the ellipse and e is the eccentricity

    b) Combing your result from a with angular momentum (L=mvr), derive an equations for the ratio of the moon's speed at periapse and apoapse in terms of eccentricty, that is vp/va.
    I ended up getting vp/va=(1+e) / (1-e)​

    c) Write down the equation which equates the total energy of the moon at apoapse with its total energy at periapse using the equation Ep=-GMm / r
    Im getting confused with this one. So the total energy would be the gravitational potential energy plus the kinetic energy right? So for the energy at periapse would look like
    E=(-GMm / r) + 1/2mv2
    the velocity being the one derived earlier.
    then using the equations derived in the first couple parts I end up getting
    E=(-2GMm + mv(1-e)) / (2a(1-e)​
    And somehow I think that is wrong. The next part asks me to show that at periapse
    v2=(GM / a)[(1+e)/(1-e)] ​
    but I may be over complicating things for part c and trying to solve d at the same time im not sure. Anyone have any ideas if I made a mistake earlier or if they can offer some advice in solving it?
  2. jcsd
  3. Jan 28, 2010 #2
    nevermind I figured it out
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