What does graph look like if velocity is discrete quantity?

Question

This sound silly question. If velocity is continuous quantity when we plot graph of falling object ( distance VS time ) it would be smooth curve and velocity VS time would be straight line but I wonder what graph look like if velocity is discrete quantity ? This is why ask this question , from this answer he said :

Perhaps the space we live in is actually discrete; i.e. if you zoom in close enough, our world is made of atomic "cells", just like a Minecraft world. Suppose each cell is a cube $1.6×10^{−45}$ meters (ten orders of magnitude below the Planck length) on an edge. We don't know if this hypothesis is true or not: what experiment would disprove it? If it were true, then some things about real numbers that we learn in math (i.e. the idea of the limit is based, that for any number you name, I can always name a smaller one*), would be "wrong" for talking about objects on that size scale.

But it would still work just as well, as an approximation, for things that we currently use calculus for -- e.g. to calculate where to aim our spaceships. The rocket equations themselves are never going to fit the situation exactly (have you accounted for that dust particle? and that one?), the numbers we put into them are never going to be measured precisely.

A model cannot be judged right or wrong in itself; only the application of a model to a real-world situation can be judged, and then only in grades -- more appropriate or less appropriate. If speed comes in discrete chunks, then there may be no moment at which the volleyball, whose arc is described by $y=−x^2$ , is ever moving at $−4$ meters/second calculus would predict at $x=2$ . Or maybe speed is continuous, and there is such a moment.

There's no way, even in principle, to tell, so we stick with the model we've got and change it only when it predicts the real world incorrectly.

So it make me curious what graph look like and how calculus can help if velocity actually is discrete chunk .

Answer 1

It is not a silly question. Velocity (and position) in any numerical computer simulation can only vary by discrete increments, determined by the finite precision of the arithmetic in use.

What happens is not difficult to describe: all quantities at a very fine scale look like piecewise constant functions defined on a fined set of isolated points. However, even in the case of the standard $32$ bit arithmetic, such discrete variations are visible only at a scale of relative variations of $10^{-7}$. Typical plots at the scale of interest cannot reveal such a discreteness of data, and one gets a perfect illusion of smoothness.

Therefore, if the real world would behave like numerical simulations at a scale of $10^{-45}$, there would be no way to detect the difference with a continuous smooth world until experiments can probe that scale. Meanwhile, the smooth, continuous model is much easier to deal with from a mathematical point of view.