Alright, so I have a few questions.(adsbygoogle = window.adsbygoogle || []).push({});

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-e^{2}) / (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 v_{p}/v_{a}.

I ended up getting v_{p}/v_{a}=(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 E_{p}=-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) + 1/2mvthe velocity being the one derived earlier.^{2}

then using the equations derived in the first couple parts I end up gettingE=(-2GMm + mv(1-e)) / (2a(1-e)And somehow I think that is wrong. The next part asks me to show that at periapse

vbut 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}=(GM / a)[(1+e)/(1-e)]

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# Homework Help: Deriving Mechanics Equations for the Moon

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