# Deriving Mechanics Equations for the Moon

1. Jan 27, 2010

### Dime

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$$\vartheta$$). 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. Jan 28, 2010

### Dime

nevermind I figured it out