[kqr]

While this reasoning may be common, I don't think it's to anyone's benefit. Talking about "the" formula for kinetic energy seems nonsensical, when there are so many ways to state that relationship. Another option is p^2=mK.

But there is a good reason to write it as K=mv^2/2 which has nothing to do with specifying a computation. It is the result of symbolic integration of p=mv with respect to v.

[a1369209993]

And, perhaps more intuitively, its the second integral of m with respect to v. This pattern also shows up in eg A = ∫∫τ∂r² for the area of a circle.