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Elliptical orbits question

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data
    An efficient way to reach the Moon is to first put the spacecraft in a low circular Earth
    orbit (radius r0, speed v0). The speed is then boosted to vp giving an elliptical orbit with
    apogee at the Moon’s orbit, ra, and perigee at r0. Show that:

    (vp/v0)^2=2ra/(r0+ra)

    2. Relevant equations

    http://en.wikipedia.org/wiki/Vis-viva_equation

    3. The attempt at a solution

    Using the Vis Viva equations, I found:
    vp^2=GM((2/r0)-(1/ra))
    v0^2=GM((2/r0)-(1/r0))=GM(1/r0)

    so (vp/v0)^2=((2/r0)-(1/ra))/(1/r0)
    Which simplifies to (2ra-r0)/ra, which isn't right.

    Where did I go wrong?
     
  2. jcsd
  3. Feb 14, 2012 #2

    D H

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    Staff Emeritus
    Science Advisor

    The vis viva equation is [itex]v^2 = \mu\left(\frac 2 r - \frac 1 a\right)[/itex], where [itex]\mu=GM[/itex] is the gravitational parameter, [itex]r[/itex] is the radial distance, and [itex]a[/itex] is the semi major axis of the orbit.

    Your mistake was using the apogee distance in lieu of the semi major axis.
     
  4. Feb 14, 2012 #3
    I thought in this case the apogee distance was the semi major axis? :-/

    If not, how do I find the semi major axis?

    Thanks for helping!
     
  5. Feb 14, 2012 #4
    Sorry, I've got it now! Thanks! :-)
     
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