newtonian mechanics - What would happen if $F=mdot{a}$?

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 $F=m*d^nx/dt^n$ ?