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It’s eminently possible that the time it takes to finish each step will still go down: half the original time, a third of the original time, etc, but that the total journey will take an infinite amount of time.
You can check this for yourself by trying to find what the series [½ + ⅓ + ¼ + ⅕ ++ …] sums to. As it turns out, the limit does not exist: this is a diverging series.
Pure mathematics alone cannot provide a satisfactory solution to the paradox, as it isn’t simply about dividing a finite thing up into an infinite number of parts, but rather about the inherently physical concept of a rate
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Rate is the amount that one quantity (distance) changes as another quantity (time) changes as well.
You can have a constant velocity (without acceleration) or a changing velocity (with acceleration).
You can have an instantaneous velocity (your velocity at one specific moment in time...
Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time.
Specifically, as asserted by Archimedes, it must take less time to complete a smaller distance jump than it does to complete a larger distance jump, and therefore if you travel a finit...
To go from her starting point to her destination, Atalanta must first travel half of the total distance.
To travel the remaining distance, she must first travel half of what’s left over. No matter how small a distance is still left, she must travel half of it, and then half of what’s still...
This is the resolution of the classical “Zeno’s paradox” as commonly stated:
The reason objects can move from one location to another (i.e., travel a finite distance) in a finite amount of time is not because their velocities are not only always finite, but because they do not change in tim...
The oldest “solution” to the paradox was done from a purely mathematical perspective. The claim admits that, sure, there might be an infinite number of jumps that you’d need to take, but each new jump gets smaller and smaller than the previous one. Therefore, as long as you could demonstrate that...
This reasoning is only good enough to show that the total distance you must travel converges to a finite value. It doesn’t tell you anything about how long it takes to reach your destination.
There's no guarantee that each of the infinite number of jumps you need to take- even to cover a fi...
How fast does something move? That’s a speed.
Add in which direction it’s moving in, and that becomes velocity.
And what’s the quantitative definition of velocity, as it relates to distance and time? It’s the overall change in distance divided by the overall change in time.
The takeaway is this:
Motion from one place to another is possible, and because of the explicit physical relationship between distance, velocity and time, we can learn exactly how motion occurs in a quantitative sense.
Yes, in order to cover the full distance from one area to anothe...
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I found this paradox is interesting, mainly because it has such a simple solution for what seems like an unsolvable problem.
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