If Obi-Wan and Anakin were paying even the slightest bit of attention at this point of the movie, they would have figured out that Palpatine can't possibly be the frail old man he pretends to be. There's no way anyone but a trained Jedi/Sith (or Chuck Norris) could have performed that swing out the lift shaft door and landed without incurring multiple fractures, punctured splanches, and other life-threatening injuries.

Transcript

Anakin: I let go.
Obi-Wan: Allow me to remind you that you're currently responsible for both my safety and that of the Chancellor of the Republic.
Anakin: Trust me.
Palpatine: You said that when you had R2 turn the ship sideways.
Anakin: And we're not dead. Yet.
Anakin: I use my grapnel to hook a beam and swing us out a door.
GM: Hmmm. Roll on Grapnel at a -8.
Anakin: ... and Force Precognition to judge the cable length so we don't slam into a wall.
GM: Okay then, at -4.
[SFX]: whoosh
[SFX]: clang!
Anakin: 17.
[SFX]: splat!
Palpatine: It's fortunate I still have Jedi reflexes; a manoeuvre like that would have broken several bones on an untrained person of my age.
Palpatine: Are you okay, Master Obi-Wan?
Obi-Wan: Yes, my bruises broke my fall.