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human
Why are so many basic Physics formulas in the form of 0.5ab^2?

The title pretty much sums up my question. To be more specific however, why are so many of the basic mechanics physics formulas all in the same general form of 0.5ab2. For example, Kinetic Energy is 0.5mv2, and spring potential is 0.5kx2. In the kinematic equation x=xit+0.5at2, the 0.5at2 shows up again. In the formula for energy of a particle in simple harmonic motion, the energy formula is once again 0.5kA2. It even shows up sometimes in some of the most basic electricity and magnetism formulas for potential energy.
Obviously some constant times x, when integrated, gives 0.5cx2, but I was curious is there is some other reason this form for equations shows up so often, or if it just a product of integration.
My apologies if this is a very basic question, or a meaningless one. I am still in high school physics and haven't reached anywhere close to a high level of understanding with regards to physics.
Thanks!
gpt
You've more or less answered your own question. The form 0.5 c x^2 is obtained by integrating the expression y =  c x. And "y = c x" expresses that one variable (y) is proportional to the other (x). Two variables being proportional to eachother is rather common, so the relation y = c x appears often and by extension the integrated expression 0.5 c x^2 also shows up regularly.

As an example, the force of a spring is given by F = k x (k the spring constant, x the length difference from the equillibrium position). We know that the potential of a compressed (or extended) spring is equal to work performed to compress (or extend) the spring. This work is determined by integral of the force (as function of distance), so the integral of F = k x, which gives us W = 0.5 k x^(2).

The other relations of this form have a similar derivation, based on the notion of two variables being proportional to eachother.