Maximal parallel velocity

[andrea1313]

TL;DR Summary

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##.