Path of a ball thrown radially from a satellite orbiting Earth

[khfrekek92]

Homework Statement

A satellite has a circular orbit 500 km above earth. A ball is thrown directly downwards radially at 20 m/s. Find the path of the ball relative to the satellite.

Homework Equations

mv^2/r=GMm/r^2

The Attempt at a Solution

From the height of the orbit you can easily solve for the velocity:
mv^2/r=GMm/r^2=>v_{satellite}=sqrt{GM/r}=7667.7 m/s

Now I can easily tell that the ball is just going to take a slightly elliptical path relative to EARTH, however I can't quite figure out (conceptually or mathematically) what this path will take relative to the satellite moving in a circular orbit.

From what I can tell, immediately when the ball is thrown downwards, it will go in reverse (slightly) relative to the satellite because it won't be able to speed up again to the original velocity of the satellite. Any thoughts?

Thanks in advance!

 

[BvU]

I would expect it to maintain tangential speed, because there's nothing to change that...
And then the ball at least initially would gain on the satellite because circumference of the trajectory is less... :-)