Xprize and tethered space station

wrote:

And that's where I'm confused. Isn't 10,000 fps at GEO a lot less
than the 25,000 FPS at LEO?

Why is it that in going from LEO to GEO you're getting rid of speed?

Maybe this will do it. Say I'm in a nice, stable, circular orbit at
GEO. I want to drop to a nice, stable, circular orbit 100 miles
lower. I don't care about the orbital period, I just want to do it.
Which way do I point my nose before I light my rocket?

Have you ever seen one of those charity things, where you drop your
penny into the slot and it starts it spinning around and around, and
keeps getting lower into the well, until at the last it is spinning way
fast just before falling into the hole?

That is exactly what's going on in space, but bigger. While you are
going around slower in GEO/close to rim of well, you have more energy
than the penny that is near the bottom of the well. You are higher.

To get the penny to go down to a lower orbit in the well, you have to
slow it down, i.e., point the rocket engine opposite the direction of
motion.