Zeno's paradox

if you do not know what is zeno's paradox, pls click here for one of the examples.

(take note, it will redirect you to one of wikipedia's page)

ok, i know i may not be the fastest to solve this paradox, but i did definitely solve it myself, without the use of external source. Here is how it goes :

Assume that the runner's velocity is 10m/s, and the tortoise's speed is 5m/s.

(which is considerably fast comparing a tortoise and a runner.)

Let's say the race is 1.6km.

and the tortoise starts 100m ahead of the runner.

so, the runner would take 10s to get to the initial tortoise starting position. The tortoise would have moved 50m in the meantime.

In phase 2, the runner would take 5s to get to the tortoise position. In the meantime, the tortoise would have moved 25m. Looking at the pattern,

Distance covered by runner and tortoise keeps halving in each phase of the 'catching up' process.

so if I were to list down the values,

it would be

I 50

II 25

III 12.5

IV 6.25

V 3.125

VI 1.5625

VII 0.78125

VIII 0.390625

IX 0.1953125

X 0.0976563

Now comes the problem solving part, which in definite terms says that if the runner cannot catch up with the tortoise, then the tortoise cannot complete the race.

Notice that phase I to X added together gives you 99.902344, which is less than 100(which is the advantage given to the tortoise, or what you can term as Phase 0)

And what does this mean?

This goes to show, that regardless of how many cycles you may want to continue on, the number would never reach 100, which is the advantage the runner gave to the tortoise.

And if you need more evidence, just add the values from the back to the top, and if u decide to stop at, let's say 6, you realize that the total value is less than the phase before 6 (which is 5).

Or that you could further the list down to 100 instead of 10, and still realize that the total value of 2 to 100 is still less than 1. This shows that this is a convergent sequence instead of a divergent one, which means that the distance covered would not pass a certain limit. Please use the limit formula to find the exact value. (I'm not providing a mathematical explanation of the paradox. )

And this is exactly the same as how 0.5 + 0.5(0.5) + 0.5(0.5)(0.5) + 0.5(0.5)(0.5)(0.5) + ... is always smaller than 1. It's the same as if u had a empty square, and you fill up half of the square, and then another half of the remaining half, and half of the remaining half... you would never fill up the square. Here's a picture to help you understand what i'm trying to put across.

.

And if you thought the square was just about to being filled up, magnify the smallest square, and you are back to square one. The square will never be filled up as long as this pattern continues for a finite number of steps, like what Zeno likes to say.

Maybe you could take the area of the original square as the distance/time required to complete the race. Hence, if even if we keep splitting time into smaller segments, this picture would illustrate that the tortoise would never complete the race, even though there is a indefinite number of cycles.

Looking at this pattern, if the runner were to give the tortoise a 800m advantage out of the 1.6km race, i believe you could tell that the tortoise would never complete the race if you were to say that the runner would never catch up with the tortoise.

The runner would take 160s to complete the race, irregardless of whether the tortoise is present. And the tortoise would complete 800m in this 160s that the runner takes to complete the race. And it brings us to...

Both of them would complete the race together!

Take note, the runner would not overtake the tortoise, and that both of them would reach the finishing line at the EXACT same time.

*don't be misguided by the limit of the distance covered by the tortoise, because the primary assumption from this limit is that time can always be broken down into smaller segment, and that the runner would not catch up with the tortoise because the tortoise would have moved a distance, no matter how little, this the next smaller time segment. Hence, if the tortoise cannot reach a certain point even if the time segment stretches out to infinity, the runner cannot reach where the tortoise position just before the moment of overtaking. Hence, if the limit of distance covered by the tortoise is 800m, the runner cannot overtake the tortoise any time/distance before 800m, and the point of overtaking is and is only at 800m.

I must say that it is purely theoretical(Zeno's paradoxes ARE, too), and that the velocity of a tortoise half of that of a runner's is unobtainable in reality.

Anyway, if you were actually convinced by this paradox that the runner can never catch up with the tortoise, even though you felt uneasy about his theory, and yet is unable to find any fault with his reasoning, you should try questioning the basic assumption made by your sub-conscious mind. That is, if the time segment would stretch to infinity, then the tortoise would definitely be able to finish the race before the racer.

I hope you understood what i'm trying to say, and feel free to clarify any doubts with me.

Cheers!