You wouldn't be able to get into orbit with a single launch. You could reach escape velocity (assuming you didn't burn up leaving the atmosphere), or a balistic trajectory.
In theory on a perfectly uniform sphere in a vacuum if you stood on a ladder, launched parallel to the ground, and knocked the ladder over on launch, you'd be in a pretty stable orbit with a periapsis of the height of the ladder.
If you didn't knock the ladder, even with a rotating sphere, eventually your orbit would coincide with the ladder again.
(I'm not sure how much frame dragging would cause your orbit to decay and there may be some other issues, like solar pressure)
To get into "stable" orbit (what does stable mean) in the real world you'd need to change your direction once you reached a certain height (typically above the majority of air). A single launch would form an orbit which intersect with your starting point (which even if you started on top of Chimborazo would still have plenty of air resistance)
I think we are just making some different assumptions. TFA also made a whole bunch of unreasonable assumptions, such as maximum time allowed on the swing, a certain height for the beam, no mention of rocket-assist at all...
When you were a kid, did you ever make the swing complete a full revolution? Not while you were sitting on it, pumping it yourself of course. But the headline and proposed question of TFA was simply "How far can you jump from a swing?" So I just made some different assumptions.
Build a swing such that the beam is about 75km off the ground or maybe just a bit more to be safe, with about 75km long suspension tethers for the seat to hang on. Get your rocket-boosted seat going at whatever velocity you need to make a full revolution plus inserting yourself into orbit at around 150km. Once your seat reaches right about the zenith of the swing's circular path, along the line running from you through the beam perpendicular to the tangent of the earth's surface below, you jump. Or "jump", since you'll mostly be upside down and puking into your fantasy tech space suit or whatever you need to get this done.
Since the seat, no longer rocket boosted, will experience a nice amount of drag once it re-enters the atmosphere, you don't even have to worry about colliding with it anymore either.
I don't see what the problem is.
By stable orbit I just mean one that does not suffer atmospheric drag, since you won't have any extra propulsion once you jump. Probably. Who knows. Do whatever you want. Reckoning with other possible sources of orbital decay is too far outside of this nonsense to bother with. :)
Even ignoring the egregiously nonsensical numbers:
> The reason is that the gravitational force of the Earth decreases with distance, so as you go higher, the force pulling you back becomes weaker. This means that your potential energy increases as you go higher.
Saying “this means that” does not make it in any respect correct.
