Why is "squaring" so prevalent in physics equations

[glee]

(bolding mine)

@Santo Rugger

So, naturally, when we take the integral of momentum, we’re going to get a squared term.

This use of ‘naturally’ is why i love this board!

[John_Mace]

@Alive At Both Ends

Can I add another related question here, rather than start my own thread? What’s the highest power to appear in an equation describing a scientific law? Are there any laws involving, say, a fourth power? A fifth? Even higher powers?

It depends. You can, to give one example, write any trig function as an infinite series, so any equation with a trig function could be written in such a way that the answer to your question is infinite. To a certain extent the way we choose to write out the equations is arbitrary-- based on the coordinate system we use or the mathematical formalism we employ.

[Askance]

@glee

(bolding mine)

@Santo Rugger

So, naturally, when we take the integral of momentum, we’re going to get a squared term.

This use of ‘naturally’ is why i love this board!

Reminds me of a story from one of those “overheard comments” books from some years ago. A student is crossing the quad in a Cambridge college and sees two dons walking towards him, one talking vigorously and ticking off points on his fingers. As they pass, the student overhears just " … and ninthly …".