Post

I think part of the difficulty is the issue of defining energy or at least characterizing something about energy. More broadly, I think it is worth considering why we care about energy to begin with. I will present two intuitive arguments that lead us to conclude the proportion $KE\propto v^{2}$ . For these arguments, I will explicitly write out what intuitive features of energy I am invoking that I think many people would accept as reasonable.

The most important key feature of energy that I invoke throughout these arguments is that energy is something that can be converted from kinetic energy to potential energy and vice-versa, and you can store potential energy. I would argue this is precisely the reason we care about energy to begin with.

The key facts about energy is that it is "conserved" and it can be "converted" between various forms of energy between wildly disparate systems. Based on these facts, we can reasonably say the following:

Additionally, I think it's worth pointing out some more assumptions that I will use:

I haven't explicitly defined kinetic energy here, but I gave a useful characterization of some features of it. There are many other assumptions that go into my argument, but I won't make them explicit for the sake of brevity. It would be an interesting exercise to see how many assumptions I haven't stated explicitly.

Scenario: Suppose we have two boxes, each of mass $m$ , moving at speed $v$ to the right with a compressed spring in-between (or anything else that could potentially separate the two boxes). This system has total energy $KE(2m, v) + U$ where $KE(\cdot, \cdot)$ is the kinetic energy and $U$ is the potential energy of the spring.