What is Potential Energy, Really? - BiteofanApple

What Is Potential Energy, Really?

July 27, 2026

As with seemingly every other man my age, I've taken up running to stay fit. Your metabolism slows as you age and what used to come for free with youth now takes time and deliberate effort to maintain. I don't care for the running, but it needs to be done.

The thing is: it does give me lots of idle time to think, just like a long walk. Both have become for me a form of meditation, a way to de-stress and to mull over complex ideas. I find that I think better when I'm moving, regardless of precisely how. Yet, near the end of every run my thoughts inevitably return to the same idea:

When will this be over? I want to be done.

Of course I want to complete my run, to go the distance I set for myself, but I also want, very badly, to do it in the most efficient way possible.

The Optimal Route

To end where you started, any route must go south for just as long as it goes north and the same goes for east and west. If you know it's three blocks west to get home and two blocks north then, because you're confined to the city grid, it doesn't matter what order you take those steps in. It's a quirk of city blocks that there are multiple shortest paths, but we'll come to that.1

E1=2U+3L<br>E2=2U+3L<br>E3=3U+3L+1D

Paths (1) and (2) take different routes but have the same total "effort" required to complete them and cover the same amount of total distance. It's only path (3) that differs. Why?

If you find that you need to add distance to your run as you go then you'll have to deviate from these ideal paths. To do that you need to use energy, more so than you would need to accomplish the ideal route. Of course you need energy to move your body in general but so long as we assume that you always intend to eventually go back home, and that your speed is constant, the energy to go from here to there is fixed by your current position along the ideal path, and thus already accounted for. The energy required to traverse that path can't change because that would require the path to get longer, and that means it wouldn't be the ideal path anymore. Thus, what can change is the amount of energy you use to get home beyond that fixed amount.

Etotal=Eideal+Eadditions

We can arrive at a formal definition by renaming a few things. If we call the shortest, ideal route a geodesic, then this added energy is defined as whatever energy is required to deviate from a geodesic. We call this "potential energy" (V). If we rename the energy required to move along the geodesic the "kinetic energy" (T) then what we have is the familiar equation from elementary physics.2

E=T+V

Crucially this potential energy is only expended to deviate from the ideal path, not to relax back toward it—that's just the process of following the new shortest path. In the (1)/(3) split in our diagram above, the energy is expended along the path of the green arrow. From there the path back is fully determined. One consequence of this is that, in our example, added blocks always come in pairs. Whatever additional city blocks you traverse away from your destination must eventually be retraced if you are to get home.

Now these are not the typical definitions of these terms that you find in elementary physics, because in that context, kinetic energy is the energy of motion. Here we've discussed it as simply the energy required to reach a given destination. However these are two sides of the same coin. Throwing a ball from A to B causes the ball to move. If we don't know where the ball will land (i.e. we don't know the value of B), then we can define the kinetic energy to be the energy of motion and use that to compute the value of B. If we know the values of A and B already, then the kinetic energy is just the energy required to reach B from A. These are the same argument given two different framings.

Keep in mind it doesn't matter that a real ball won't end its journey mid-air, but it doesn't end it's journey when it hits the ground either. It's still riding on the moving Earth.

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One obvious complaint you may have about my city-block example is that it's wrong. The ideal path between my starting point and ending point is a straight line! That's Euclid's first postulate from geometry! It's the very definition of what a straight line even is, right?

Isn't all this just a bit silly?

And you'd be entirely correct, kind of.

The thing is, you can't run as the crow flies, at least I can't. Confined to the ground, such a straight line path isn't an option. This is an important restriction because it allows us to redefine what the word distance even means. We will come out of that discussion with something much more powerful than we might first expect. To do that I have one question we must answer:

What's the distance from Rome to Edinburgh?

A clever reader might want to know which units I want them to use, but the...