[NaiveBayesian]

I agree that this feels intuitive, that potential energy should increase linearly with height.

But in the end, it's all up to the units/quantities we choose to measure, no? If we, say, decided to measure "Squenergy" in Sqoules, with 1Sq² = 1J, then suddenly, squenergy does increase linearly with speed! The formula for kinetic Squenergy becomes sqrt(m/2)v.

Of course this complicates other stuff, like potential Squenergy becoming sqrt(MgH), it not being additive, etc.

[Matumio]

No, it's not an arbitrary choice.

You can convert any energy form into thermal energy. When you boil water with an immersion heater, what happens if you use two heaters? Right, they add up. But not if you think in Squenergy. Or you can measure the effect (in heat) that a ball impact has. In Squenergy two balls don't add up add up, so it's less natural to think that way.

[feoren]

The difference is that Squenergy is not conserved.