Here's an proposal on how to get from point A to point B in zero-gravity without using any propellant and the question why it wouldn't work:
A closed tube, filled with water and a round (solid) object. If you need equations, the volume of enclosed water is the same as the round object, but the round object is 10 times lighter. (imagine a glass with water plus a ping-pong ball).
here's a picture:
On earth the round object will float on the water inside the tube (subject to one G).
In zero-gravity the round object has no preferred position.
If we accelerate the tube in zero-gravity by one G, the situation is the same as on earth, the round object "floats". In this example we are accelerating the tube from the left side to the right side. The round object will consider floating to the right AS LONG AS THERE IS acceleration.
Now consider adding a pipe to the bottom of the tube and connecting it to the top, a loop. Inside the pipe is a small water pump.
If we give this apparatus a push, say one G in a zero-g environment, the round object will move "up", but now we start the water pump and spray the water on the round object, we try to submerge it. It will resist and impart a impulse on the water. like trying to hose down a air balloon floating on the pool. The pump will feel a resistance and hence the whole apparatus will move.
Just running the pump at constant speed, same volume of water per second, will do nothing. but if we run the pump faster and faster the whole apparatus will start moving:
the amount needs to be geometrical. The point is that we need to keep the apparatus feeling an acceleration, since only then will the round-object "float" and resist the incoming water at the top, hence we have something to "push on".
Before you blow the "foul whistle": consider the situation if there were no round floating ball in the apparatus.
(the pump runs on solar power or pre-charged battery)
(disclaimer: i know standing on a sailboat and blowing into the sail will not get me anywhere, action<->reaction)
thanks a bunch Sklivvz for the edit. sometimes the idea just needs to get out, never the mind how it looks like : P