TITLE: How does anything move?
QUESTION [16 upvotes]: 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?

REPLY [4 votes]: Who says all derivatives have to change continuosly? With a suitable setup, acceleration can jump from zero to something. Location and velocity can not change discontinuously, but acceleration can.
No, there are no "infinitely many derivatives of velocity".
In your question, you write "acceleration has to increase from zero and therefore the third derivative of motion with respect to time has to be positive." This assumes that there is a third derivative, which implies that the second derivative is continuous. But the acceleration, the second derivative, may not be continuous. If it is not, there is no defined third derivative, the assumption is false and thus the conclusion does not hold.
Regarding Zeno's Paradox, I would not compare it with this. The paradox arises when time is not taken into account. In my view, once you take time into account and speak of derivatives, it is no longer Zeno's paradox.
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