how to intanteneous proof by mathematically in Physics
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Answer:
Read the explanation
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Explanation:
The fastest human in the world, according to the Ancient Greek legend, was the heroine Atalanta. Although she was a famous huntress who even joined Jason and the Argonauts in the search for the golden fleece, she was renowned for her speed, as no one could defeat her in a fair footrace. But 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.
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: Zeno’s paradox. Here’s the unintuitive resolution.
A scuplture of Atalanta, the fastest person in the world, running in a race. If not for the trickery of Aphrodite and the allure of the three golden apples, nobody could have defeated Atalanta in a fair footrace.
A scuplture of Atalanta, the fastest person in the[+]JEBULON / WIKIMEDIA COMMONS
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 that each new jump got smaller and smaller than the prior 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.
For example, if the total journey is defined to be 1 unit (whatever that unit is), then you could get there by adding half after half after half, etc. The series ½ + ¼ + ⅛ + … does indeed converge to 1, so that you wind up covering the entire needed distance if you add an infinite number of terms. You can prove this, cleverly, by subtracting the entire series from double the entire series as follows:
(series) = ½ + ¼ + ⅛ + …
2 * (series) = 1 + ½ + ¼ + ⅛ + …
Therefore, [2 * (series) - (series)] = 1 + (½ + ¼ + ⅛ + …) – (½ + ¼ + ⅛ + …) = 1.
Simple, straightforward, and compelling, right?
By continuously halving a quantity, you can show that the sum of each successive half leads to a convergent series: one entire “thing” can be obtained by summing up one half plus one fourth plus one eighth, etc.
By continuously halving a quantity, you can show[+]PUBLIC DOMAIN IMAGE
But it’s also flawed. This mathematical line of 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 you to reach your destination, and that’s the tricky part of the paradox.
How could time come into play to ruin this mathematically elegant and compelling “solution” to Zeno’s paradox?
Because 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. Under this line of thinking, it may still be impossible for Atalanta to reach her destination.
One of the many representations (and formulations) of Zeno of Elea’s paradox relating to the impossibility of motion. It was only through a physical understanding of distance, time, and their relationship that this paradox was resolved.
One of the many representations (and formulations)[+]MARTIN GRANDJEAN / WIKIMEDIA COMMONS
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.