### Pujjle yagain

There is a very famous puzzle known as Zeno's paradox. It has remained one of my favourites thru the last few years - specially because of its simplicity. This is how it goes.

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Suppose there is a guy, X, who is 10 metres in front of another guy Y. Further suppose that X and Y are running in the same direction; Y running 10 times as fast as X. Then,

when Y covers 10 metres, X will have covered one metre (so, he is one metre in front of Y). When Y covers that 1 metre, X would have moved a further 0.1 metres. When Y covers the 0.1 metres, X would have moved 0.01 metres ahead... till infinity.

If this goes on till infinity, how will Y ever catch up with X? Yet, we know thru common sense, that given the above initial conditions, Y will definitely overtake X. That too, pretty quickly!

That is the paradox:). Now, obviously, Zeno's reasoning was faulty. Where and what is the error?

*****

Suppose there is a guy, X, who is 10 metres in front of another guy Y. Further suppose that X and Y are running in the same direction; Y running 10 times as fast as X. Then,

when Y covers 10 metres, X will have covered one metre (so, he is one metre in front of Y). When Y covers that 1 metre, X would have moved a further 0.1 metres. When Y covers the 0.1 metres, X would have moved 0.01 metres ahead... till infinity.

If this goes on till infinity, how will Y ever catch up with X? Yet, we know thru common sense, that given the above initial conditions, Y will definitely overtake X. That too, pretty quickly!

That is the paradox:). Now, obviously, Zeno's reasoning was faulty. Where and what is the error?

## 2 Comments:

Say speed of X is v m/sec. Then speed of Y is 10v m/sec. By simple mathematics, it can be found that Y will cross X in time 10v/9 seconds. So Y will not cross X before 10v/9 sec.

Now for the paradox:

In 1/v secs, X crosses 1m and Y crosses 10m. Y takes 0.1/v secs to cross that 1m , in which X crosses 0.1m. Again, Y takes 0.01/v secs to cross that 0.1m , in which X crosses 0.01m. This goes on till infinity.

Now if one adds up the time till infinity we have the total time = (1/v + 0.1/v + 0.01/v + 0.001/v + ....), which, when limit tends to infinity, will equal to 10v/9 seconds. This is in accordance to the above result that Y will not cross X till 10v/9 sec, but after that.

So Y does not cross X till 'infinity', only after that. (Here, infinity refers to infinity in above series and not in actual time).

Can't explain any better.

perfect, manka.

What Zeno didn't know was that an infinite (geometric) series can have a finite sum. He assumed it would be unbounded.

as we know now, an infinite gemoetric series is unbounded only if the multiplication factor is more-than-or-equal-to one.

This paradox was solved only after there was a proper understanding of infinite geometric series (which was in the second half of this millennium - i still believe that we are in the 20th century;))

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