Working out how much force I would need to give the ship an acceleration of .5 meters per second (about 1/20 of Earth's gravity), it came out to 150,000 newtons.  Now I needed to know how much power that is.  A quick look on the web tells me that One watt is One Newton meter per second. Hey, this is easy.  One watt equals one newton.  I have megawatt reactors on board and I only need 150,000 watts.

Then Peruraptor pointed out to me, as the reaction mass used to generate that newton  goes down, the amount of energy needed goes up.  I was using one kilogram of mass per second, which gives a power need in the tens of gigawatts.  Quite a bit more than the megawatt range reactors on the ship put out.  While the folks on the forum came up with some interesting ideas of how this could be worked around, not all power goes into thrust.  A lot of it is waste heat, and it's not easy to get rid of gigawatts of heat in a vacuum.  I would be trading one cheat for another.  Best to come clean, show where the error was made, and admit that the trip can't actually be made in three weeks with the current design of the ship and some suspension of disbelief is necessary.  This is why NASA uses real engineers instead of cartoonists for their rocket equations.

Peruraptor also mentioned "If you want to know about a really power hungry engine, check out the photon drive. No reaction mass required, but each Newton needs 300MWs of power!"

Doug Lampert explained it quite well as: "Now, what are you using for reaction mass and how fast is it going? Let's say we're using 1 kg/s of reaction mass (as in the error explanation page). Ignoring relativity (which is largely irrelevant for this), we need that reaction mass to be travelling at 150,000 m/s (relative to the ship) to get that thrust (simple conservation of momentum done as a rate calculation). And THAT takes 11.25 gigawatts (simple 1/2 m v^2)."