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In this other question I have found the following:

A space elevator can climb at a low, constant vertical speed (albeit for a very long climb), and gets its orbital speed for free [borrowed from Earth's rotational speed of course].

So, does this this mean that to start the elevator, we invest some energy E, and then from some point elevator will start moving itself and give us E back, maybe even E'>E?

(if you think this is more for Physics SE I'm fine with that)

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    $\begingroup$ Have you seen Tom Spilker's answer to that question, which goes into considerable detail on that point? $\endgroup$ – Hobbes Sep 11 '18 at 8:26
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Yes, but E' > E is dubious, at least in case of Earth. The point of reversal lies exactly at the GEO altitude, where the elevator moves at speed proper for that altitude and doesn't put any strain on the tether. As it climbs further, past GEO its 'horizontal' velocity exceeds orbital velocity of that altitude, and it's pulled away from Earth, needing to brake against the tether and putting a strain on it.

The problem here is that you'd need a lot of that tether extended past GEO to get E' > E and that tether is not cheap. It's much easier to extend, say, 10% of its length past GEO and put an asteroid or a big, sun-shaded chunk of ice on the end, to offset the mass of the tether pulling towards Earth.

But in short, yes, if you could have the tether sufficiently strong and light (...not to cause Earth to wobble in its axis...) you could extend it pretty much indefinitely and give your payload pretty much arbitrary speed (although if you start entering relativistic velocities, atomic bonds begin to misbehave...) - and in case of Earth you're obviously limited by the Moon, which would collide with the elevator crossing its orbit.

OTOH you shouldn't need to worry about energy too much - the elevator will be a powerful electrodynamic tether producing lots of energy from Earth magnetic field.

(although it seems you'd need to give 13 kWh to lift a kilogram of mass to GEO using the elevator - and then you'd get back 103 kWh if the tether stopped just shy of the Moon orbit.)

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