1. quick proof
Rockets show that energy is transformed in proportion to force times time (ft), which equals mass times velocity (mv) for an accelerating mass. Rockets usually burn fuel at a constant rate and produce a constant force. It means that a defined amount of fuel produces a consistent amount of force times time. Rockets do not produce a consistent amount of force times distance, which would equal mass times velocity squared. Rockets do not see distance. Distance relates to the starting point, and it varies with velocities, which vary with reference frames. In other words, a rocket will have a different velocity relative to the earth and to the moon. With different velocities, there will be different distances to the starting point.
Physicists use a distracting irrelevancy to counter these facts. They show mathematics for the power of a rocket which is independent of reference frames. To do this, they relate power (rate of energy addition) to the separation velocity of the exhaust, which is always constant independent of reference frames. They then say equations balance, so there is nothing wrong with the definition of energy.
Balancing equations in that matter is a distraction, which does not directly test the definition of energy. But they indirectly prove the definition of energy wrong in doing so. When they use the separation velocity of the exhaust as the reference for rate of energy use, they throw out the distance to the starting point. If the starting point is not relevant to their argument, then energy is not transformed in proportion to force times distance.
To avoid the irrelevant distraction of rocket power analysis, the proof uses the force of the rocket only. The force of a rocket is independent of the definition of energy. The force can then be used to replace gravity in the falling object issue, while the burn time of the rocket shows how much energy is used. This test shows that the original basis for the definition of kinetic energy was wrong in starting with the force-distance analysis. (See History.) The force-time analysis should have been the basis for the definition of kinetic energy.