What would happen if $F=m\\dot{a}$?

Question

What would happen if instead of $F=m*d^2x/dt^2$, we had $F=m*d^3x/dt^3$ or higher? Intuitively, I have always seen a justification for $\\sim 1/r^2$ forces as the "forces being divided equally over the area of a sphere of radius $r$". But why $n=2$ in $F=m*d^nx/dt^n$ ?

Answer 1

Without Newtons 2nd Law in its correct form, then momentum would not be conserved. Conservation of Momentum is a direct consequence of combining Newton's 2nd and 3rd Laws.