I managed to confuse myself with a simple thought experiment. I have a space ship that uses mechanical springs loaded with balls for thrust. We start in a reference frame O where the ship isn't moving and fire one spring, which gives the ship a velocity v. Pick a new frame O', moving at v in the same direction, so the ship appears static again and fire another spring. If the mass of each ball is small in comparison to the ship, every repetition of this is the same, so the ship appears to increase its velocity by v each time a spring is fired. This doesn't make sense though, because let's say I fired 10 springs, then by tracing back to the original frame O the ship would appear to be moving at 10v. However, each spring is identical so it should store the same energy. Obviously E=M/2 * (10v)^2 != M/2 * 10 * v^2.