[SyzygyRhythm]

Cheat answer: velocity is a vector, and can be negative, while KE is a scalar and has to be positive. Therefore you have to square v to get rid of the minus sign.

Why not take the absolute value? Nature hates those, probably because the derivative is undefined at 0. So squaring it is.

[signa11]

why not raise to any other even power ?

[meowkit]

One way of thinking about that is higher order even powers just reduce down to two.

For the purpose of inverting a negative vector, you can think of squaring as rotating the vector around the unit circle, 180 degrees, to make it positive. Higher order powers just keep rotating that vector back and forth- from this perspective the other even powers are the same transformation. Obviously with the magnitude being different.

[qzw]

That doesn’t answer the title question of why it’s quadratic wrt speed.

[SyzygyRhythm]

To get speed from velocity, you need a square root, which is also awful (for the same reason that abs is awful).