The fastest human in the world, according to the Ancient Greek legend, was <a href="https://en.wikipedia.org/wiki/Atalanta">the heroine Atalanta</a>. Although she was a famous huntress who joined Jason and the Argonauts in the search for the golden fleece, she was renowned for her speed. No one could defeat her in a fair footrace. She was also the inspiration for the first of many similar paradoxes put forth by the ancient philosopher Zeno of Elea about how motion, logically, should be impossible.

Background 

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

Only, This Is Flawed Too

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.

What Is Velocity?

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 time unless acted upon by an outside force. If you take a person like Atalanta moving at a constant speed, she will cover any distance in an amount of time put forth by the equation that relates distance to velocity.

How Does Rate Provide A Solution

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 finite distance, it must take you only a finite amount of time. And therefore, if that’s true, Atalanta can finally reach her destination and complete her journey.

Including The Concept Of Time

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 remaining, and so on, ad infinitum. With an infinite number of steps required to get there, clearly she can never complete the journey. And hence, Zeno states, motion is impossible: <a href="https://en.wikipedia.org/wiki/Zeno%27s_paradoxes">Zeno’s paradox</a>. Here’s the unintuitive resolution.

So, What Is Zeno’s Paradox?

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) or an average velocity (your velocity over a certain part or whole of a journey).But if something is in constant motion, the relationship between distance, velocity, and time becomes very simple: distance = velocity * time.

What Then Is Rate?

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 finite distance-occurs in a finite amount of time. If each jump took the same amount of time, for example, regardless of the distance traveled, it would take an infinite amount of time to cover whatever tiny fraction-of-the-journey remains. This way, it is still impossible for Atalanta to reach her destination

Why Is The Initial Solution Flawed?

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 the total sum of every jump you need to take adds up to a finite value, it doesn’t matter how many chunks you divide it into. Simple, straightforward, and compelling, right?

The Initial Solution To The Paradox?

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 another, you have to first cover half that distance, then half the remaining distance, then half, etc.But the time it takes to do so also halves, so motion over a finite distance always takes a finite amount of time for any object in motion. This is still an interesting exercise for mathematicians and philosophers.

Final Thoughts 

I found this paradox is interesting, mainly because it has such a simple solution for what seems like an unsolvable problem. 

Travel half the distance to your destination, and there's always another half to go. Despite Zeno's Paradox, you always arrive right on time.

What Then Is Rate?

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