Sounds like something is loose if the movement of your car effects your amp. It could be that your power wire is not connected properly at the amp or battery, remote wire is not connected properly at the amp or the deck, loose ground, or the amp is just fucked up.
sounds like you dont have your amp bolted down, and yea it sounds like the power wire is loose becuase if it was the ground i think you would have issues with it turning itself back on...it has nothing to do with the gas if you dont believe me just leave the car sit in park and hit the gas and see if it does it!! I bet it dont!! Try playing with the wires on the amp and i bet youll be surprised as to what you find out.