So by the Law of Conservation of Momentum, we know that m1v1 = m2v2, which makes sense with momentum remaining constant, and is used for rockets and the like, changing mass by removing parts of itself.
But, this seems to break the Conservation of Energy (KE + PE = C), because, for instance, if you lose half of your mass, you would move double your previous velocity (and to those that are pedantic, yes, I know it refers to a body that is already in motion in a closed system). In the case of Momentum alone this works, but this would cause Kinetic Energy to increase by a multiple of 2, without pulling from Potential Energy as far as I can tell.
Let us give the example of a hypothetical rocket of 20kg, moving at 30 m/s. This would make for 9000 Joules. If removing a stage makes it 10kg, it would move at 60 m/s to conserve momentum, but in doing so would gain 9000 Joules of energy from seemingly nowhere, becoming 18,000 Joules.
So how does Conservation of Momentum not contradict the Conservation of Energy? What am I missing that allows these two seemingly contradictory laws to coexist?