E.g. how can the car accelerate from rest to 5m/s if it first has to be going at 10^(-100) m/s which in turn requires it to have gone through 10^(-1000) m/s, etc.? That is, if a car is going at a speed of 5m/s, doesn’t that mean the magnitude of its speed has gone through all numbers in the interval [0,5], meaning it’s gone through all the numbers in [0,10^(-100000) ], etc.? How can it do that in a finite amount of time?
In: Mathematics
This is just Zeno’s Paradox, but with velocity instead of displacement. Yes, you can perform an infinite amount of tasks, as long as you define the sum time of those tasks to be convergent. Just as Zeno’s arrow performs an infinite amount of tasks, so too does the car. It’s fine because you’ve defined those tasks to take place within a finite amount of time.
Are you familiar with [Zeno’s Paradox](https://en.wikipedia.org/wiki/Zeno’s_paradoxes)?
Read through that a bit and you should be able to see how the same logic applies to your velocity example, and therefore why your example is no more a paradox than Zeno’s.
Step 1) Stand up
Step 2) Take step
You just moved maybe a foot, but to do that you had to cover a near infinite amount of space.
Just because there is an infinite amount something doesn’t mean you can’t go thru them. So yes, the car’s velocity went from 0 to say 5m/s and thus had to cover every possible velocity in-between in only a few seconds.
To quote YouTuber John Green (in his book A Fault in Our Stars):
> There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities
What’s stopping it, in your mind? In other words, can you explain why passing through an infinite number of values in a finite amount of time is somehow a problem?
Your question leads to a fun joke (which I am sure I will butcher).
A mathematician and an engineer are having a discussion. The mathematician says:
> We shouldn’t exists! To procreate, you must able to touch your partner. So you walk half way there. Then you walk half of the additional distance. Then again! And so on and so forth. Therefore we can’t procreate!
The engineer’s answer:
> I can get close enough
The easiest answer is that the universe isn’t infinitely divisible. Quantum physics means there is a minimum amount of size, distance, time, and energy. One you divide down to that minimum number you must stop dividing.
So let’s say you need to move 256 of these units. You first need to move 128 of these units. But you must first move 64 of those units, or 32, or 16, or 8, or 4, or 2, or 1. I’ve you are down to one you successfully move the one, then the next one, etc.
Planck time, the smallest unit of time, is 10^-44 seconds. Which is roughly 10^-35.
So in the first plank time it would move 1 planck distance, in the second planck time it would move 3 planck distance, etc.
You may ask “how does it cross the space”? The scar is that it doesn’t, things basically teleport at the smallest level. In reality it’s more complex like that as we are all clouds of probability and nothing actually exists in any specific place, but teleporting planck distances in planck time is the most comprehensive explanation.
Also, we invented calculus to mathematically solve the problem of infinite series in finite containers.
It doesn’t have to go through all of those infinite fractions it leaps through like stepping stones, going from one to two doesn’t mean you have to stop at any of the numbers between.
The same way you can stand up through an infinite range of positions. There are an infinite set of numbers between 0 and 1, and between 1 and 2, like 0.5 and 1.333. That does not stop you from counting your fingers 1, 2, 3, 4, 5. Numbers are a tool humans use to describe nature, but nature doesn’t need numbers to work.
From the way you phrased your question you have some more advanced math knowledge, so I’ll add this: You asked how you can move through an infinite set of possible velocities in finite time. Let’s take the simple case of constant acceleration a over a time t resulting in a velocity v starting from rest, so v=a*t. So if you have an acceleration of 1m/s² for one second you get v=1*1=1m/s. Now, what happens if we look at the same period of time but split it into two steps, same constant acceleration as before. Part 1 is the first step and part 2 is the second step, and the final velocity is vf. So vf=v1+v2, and v1=a*t1, and v2=a*t2, and the total time t=t1+t2. So you get vf=a*t1+a*t2=a*(t1+t2)=a*t, back to where we started. You can do this infinitely many times, break the acceleration into as many slices as you want, but each time you are decreasing the time of each slice by the same ratio that you are making more slices, so it cancels out. The only thing that changes is how small of a slice you consider or bother to look at, it doesn’t change the result.
Because you can’t divide the difference into an infinite number of divisions. You can always divide again, but every division you do will always result in a finite number of divisions. At no point no matter how many times you divide will you reach a result of an infinite amount of divisions. Your ability to divide is limitless but the number of divisions you create are finite. I e. You can never just “divide one more time” and go from a finite number of divisions to an infinite number of divisions.
>That is, if a car is going at a speed of 5m/s, doesn’t that mean the magnitude of its speed has gone through all numbers in the interval [0,5], meaning it’s gone through all the numbers in [0,10-100000 ], etc.?
Kinda, yes, it did. If you could stop time and divide it as you wish, watch it accelerate frame by frame.
>How can it do that in a finite amount of time?
What do you mean ?
Imagine a slower interval of time if it helps you. Take a tree that grows 1 meter in a year, measure it every 0,1 second that passes for a year. You would have a pretty detailed graphic of it’s growth speed.
Why does something similar at higher speed bother you ?
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