kinematics - How does anything move?

So in order for two things $A$ and $B$ to move apart, for example, relative to each other, $B$ can be set into motion away from $A$. This means that we have to increase $B$'s velocity and therefore the acceleration has to be positive. If the acceleration has to be positive and $A$ and $B$ were stationary before, that means that $B$'s acceleration has to increase from zero and therefore the third derivative of motion with respect to time has to be positive. The third derivative has to increase from zero and etc. etc. Is this just another way to state Zeno's paradox (and thus a dumb question) or does motion really involve an increase of the magnitude of infinitely many derivatives of velocity?