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
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.
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