I learned in Grade 11 Physics that energy is conserved. That fact depends on the truth of the following fact. For any system of objects, if its mass and momentum remain unchanged in one frame of reference, then the change in Kinetic energy is the same in all frames of reference. I define a frame of reference in such a way that for each constant velocity and orientation you could travel at, there's one frame of reference and in that frame of reference you're at a specific orientation with zero velocity. How do I prove that when the total mass and momentum of the system is unchanged, the change in kinetic energy is the same in all frames of reference? Could this be the solution?
It's trivial to show that when the mass and momentum don't change in one frame of reference, they don't change in any frame of reference and the momentum in one frame of reference can be completely determined by the mass and momentum in another frame of reference. It's also easy to show that for any system, rotating it doesn't change its kinetic energy and therefore that rotating it doesn't change its change in kinetic energy. All I have left to show is that for any system whose mass and momentum don't change, the change in kinetic energy will be the same for that system and a version of it that's moving along at constant velocity at the same orientation. I will show that as follows.
The kinetic energy of each object can be expressed as a sum of the kinetic energies of each component if its velocity so the total kinetic energy can also be expressed as the sum of the total kinetic energies of each component. The math shows that the kinetic energy of each component of the system is the sum of the kinetic energy of that component of an object with the mass of the system and the velocity of its center of gravity and the kinetic energy of that component of the system in the frame of reference of its center of gravity. That means the kinetic energy of any system is the sum of the kinetic energy of a single object with the same mass and momentum and the kinetic energy of the system in the frame of reference of its center of gravity so if its total mass and momentum don't change, its change in kinetic energy is the same in all frames of reference.