What is the Maximal Parallel Velocity of a Satellite in an Elliptical Orbit?

[andrea1313]

TL;DR

maxime velocity

so I have this problem from olympiad
the maximum value of the satellite velocity component parallel to the high axis of the ellipse(a)
and the answer should be vmax=a/b*sqrt(G*M/a). How to prove this?

 

[tnich]

Here is an approach:
1) Write the equation of the ellipse with the Earth at the origin and the semi-major axis along the $x$-axis. Note that $|\dot x|$ is at a maximum when $x=0$. (Why?)
2) Find the satellite acceleration in the $x$-direction when $y=0$.
3) Differentiate the equation of the ellipse twice with respect to time to get an equation relating $x,\dot x, \ddot x, y, \dot y$, and $\ddot y$. Solve this equation for $\dot y$ when $y=0$.
4) Find $y$ when $x=0$ and use conservation of angular momentum to find $\dot x$ when $x=0$.