[lxmorj]

The asteroid is the reaction mass. Send a solar collector and a rail gun sled that fires bits of asteroid really fast backwards.

[rtkwe]

That's a massively complicated endeavor and would require a lot of knowledge we don't have at the moment principally zero G mining and remote robotic assembly (time lags will mean it'll have to be very automated).

[Aeolun]

That sounds like experience we’ll never get if we do not even make the attempt.

[rtkwe]

Eventually, it's not the first step you take though.

[mLuby]

Forget railguns; an automated burger flipper could launch chunks of rock at escape velocity.

[rtkwe]

The faster you can accelerate them the more dV you get out of each chunk so you want to get them moving as fast as possible. If you're barely tossing them off the asteroid you might need to throw away a majority of the asteroid's mass to move it into Earth's orbit.

[gpm]

And reliable railguns.