Why is the energy of motion proportional to the square of the velocity?

Based on a geometric demonstration by Johann Bernoulli, we show that the principles underlying the work-energy theorem are more fundamental than those discussed in the standard derivations. By introducing the notion of a mechanical Bernoulli process—a systematic process for destroying velocity and measuring work—we derive the relationship between the amount of work and the square of the velocity solely from the superposition of motion and the law of inertia.