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

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.

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.

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

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.