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I have a question that has been bugging me since last night. Let's say we have a car travelling at $5$ms$^{-1}$ at one instant. From $5$ms$^{-1}$ it accelerates uniformly to $6$ms$^{-1}$ in $2$ seconds. How do I wrap my head around the fact that the car had infinite number of velocity values/states during the 2 second interval of time while it transition from $5$ms$^{-1}$ to $6$ms$^{-1}$?

Khant Rain
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    Forget about speed, acceleration, all that - over the course of $2$ seconds, we experience infinitely many time values/states. This is just a general feature of how the "continuous world" behaves: our intuition about each state taking up some nonzero amount of "room" isn't applicable, and we can indeed fit infinitely many "things" into a "finite box." (Conversely, if we take the stance that time is appropriately discrete then this issue doesn't even arise: there are only finitely many moments in that $2$ second interval, according to this picture, so there's nothing to say.) – Noah Schweber Jul 18 '20 at 06:01
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    You've already wrapped your head around the fact that the car occupied an infinite number of positions and it's only the infinity of velocities that bothers you? Why is that? And why is this a problem in mathematics rather than physics or philosophy? – bof Jul 18 '20 at 06:02
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    This is the point of the Zeno paradox where Ahilles must make an infinite number of moves to catch up with the turtoise. The paradox is resolved by the Heisenberg Uncertainty Principle in Quantum Mechanics stating the position and momentum (velocity) are not precisely defined due to the wave properties of matter. – Hiroyashu Jul 18 '20 at 06:09

2 Answers2

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The thing you are referring to is the "Intermediate Value Theorem" that holds for continuous functions. It says if ${f}$ is continuous on ${[a,b]}$, then ${\forall\ \phi \in \mathbb{R}}$ such that ${f(a) < \phi < f(b)}$ there exists a ${c \in (a,b)}$ such that ${f(c)=\phi}$. In other words, all "in-between" values are taken on at some point by the function.

Now when it comes to the Mathematics describing the car - there should be no confusion about it taking on an (uncountably) infinite number of values during the acceleration if the function describing it is piecewise continuous. It makes perfect sense (the proof for the IVT is quite simple thankfully!).

So your question is Fundamentally rooted in Physics or Philosophy. If it turns out that our 3 dimensional space is actually discrete (that is, there is such a thing as a "smallest possible length") - then the Mathematics we do is just a continuous approximation of a discrete process - and while the number of velocities the car takes on will likely be monstrously huge - it will not be infinite. On the other hand - if space is not discrete, then provided that there are no "big forces" at play which force it to take on discrete points in space - then both it's position and it's velocity really do take on an infinite number of values in-between. Which I don't find particularly weird - no weirder than time itself being a continuous process, anyway.

Riemann'sPointyNose
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A Mathematician's response:

It just does.

A Physicist's response:

It didn't. Fundamentalist's are still working out the details on if time is quantized or not. If it is, that means two things.

  1. Einstein was wrong
  2. There are a finite amount of time-slices between any two intervals. Which mean's the car actually went through many many (but still finitely many) velocities during it's acceleration.

Even if time isn't quantized, atoms still are. Which means that when the car accelerated, the combustion in the car was transmitting quanta of energy to it's wheels, which means there was a certain $N$ number of molecules that combusted, each contributing to the final velocity. Hence, it still went through a finite amount of different velocities.

A Philosopher's response:

Nothing is for certain.

Graviton
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