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Zeno paradox 2 Achilles and tortoise

2022-07-19 02:05:00 【sun_ xo】

I saw Wu Jun's article about Zeno and his paradox two days ago , About the second paradox , It is said that Aristotle was once baffled , This reminds me of a past event ：
This paradox can be described in this way ：
Ancient Greek hero Achilles ran a race with the tortoise , The tortoise's speed is v, Achilles' speed is turtle's m times , Let the tortoise run a distance first d After the game starts , When Achilles advances a distance d after ,
At the same time, the tortoise moves forward d/m, When Achilles advances again d/m after , At the same time, the tortoise moved forward again d/m^2, And so on , Because distance can be subdivided infinitely , As a result, Achilles will never catch up with the tortoise .
At first glance, this question seems very absurd , It obviously goes against our common sense , Even primary school students know that this is a simple pursuit problem , The solution is as follows ：　
　　1) m*v*t = d+v*t
　　2) m*v*t - v*t = d
　　3) (m-1)*v*t = d
　　4) t = d/((m-1)*v)
final t It's the time it takes Achilles to catch up with the tortoise .
Although the solution itself is no problem , But it is by no means a positive solution , I think Mr. Zeno knows something under the spring , I'm sure I won't be convinced . I once chatted about this topic in a small chat , As a result, there was a buddy
Immediately point out that this is a problem of summing infinite decimals , And give a positive solution to this paradox on the spot ：
set up s Equal to Achilles catching up with the tortoise after the start of the game （ If we can catch up ） The distance the tortoise traveled ,t The time it takes for Achilles to catch up with the tortoise , be ：
　　1) s = d/m + d/m^2 + d/m^3 + ... + d/m^n
　　2) s/d = 1/m + 1/m^2 + 1/m^3 + ... + d/m^n
　　3) m*s/d = 1 + 1/m + 1/m^2 + 1/m^3 + ... + d/m^n
　　4) m*s/d - s/d = 1
　　5) s*(m-1)/d = 1
　　6) s = d/(m-1)
　　7) t = s/v = d/((m-1)*v)
This is the positive solution , In fact, there is no profound mathematical knowledge , The key is 2 and 3 Transformation between , These two formulas are subtracted to get the 4 Step , Quite wonderful . The final answer and the pursuit of the problem
The answer is the same . Zeno must have nothing to say this time .
By the way , My friend is not a great Xueba , It was just that he had just graduated from college , Remember this knowledge . When he writes the answer , On the one hand, I admire very much , On the one hand, it also seems
Once known feeling , Maybe I have learned these before , But it has already been returned to the school , No wonder people often say that the peak of a person's intelligence is his high school and college years .
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