[emilfihlman]

You don't need to hold the orbit. Just jump to the elevator and grab on.

[t_fatus]

Would be nice to get the savings in delta-v, I've no idea how to compute this.

[marcosdumay]

My estimate is ~9.5km/s from the Equator to the hook. Add some more for atmospheric losses.

[s1artibartfast]

you can use the partial escape velocity equation. ~10 km/s to get to GEO without the radial velocity. delta-V to the moon is about 18 km/s (one way). Keep in mind that kinetic energy is proportional to velocity squared.

[emilfihlman]

A ballistic approach is enough so the kinematics are simple, too.

[extropy]

The gravity is only 3% up there, so more tricky. And reducing with square of the distance.

Also every second you are accelerating straight up costs you gravity worth of Delta-v because gravity losses. So need to be quite short and intense burn to be worth it.

[emilfihlman]

Eh?

You don't need to get sideways velocity at all. It's enough to only reach the distance. This is a major and huge saving in fuel.