Rabbit and Tortoise..

Folks..

Not too sure, whether this question has been asked earlier or not. If so, I am really sorry.

Let us now go back to the old Tortoise (X) and Rabbit (Y) story. Say the speed of Y is 10 times X. And alos X has gotta initial lead of 100 units. And lastly let us assume speed of X is 1 unit/sec.

Time taken for Y to cover 100 Units is 10 sec. By this time X wud have covered 10 units.

Now Time taken for Y to cover 10 Units is 1 sec. By this time X wud have covered 1 unit.

Again...wont this iteration go on forever ?. Does this mean that Y will never be able to catch up with X ?. We all know that'z not correct.

What'z wrong with my approach ??.

thanks,

Actually, by introducing time into your description of the problem, you have made the paradox abundantly clear.

The first iteration is 10 seconds, the next is 1 second, the next is 0.1 seconds, and so on. So it is clear that the analysis applies only for 11.111... seconds. After this time, the rabbit passes the

Nothing that I see, except your conclusion that Y will never catch X. You stopped less than a second too soon. In 12 seconds Y will have gone 120 units and X only 112, so you can easily see that Y passes X between 11 and 12 seconds after the start (exact time not needed for this post so time wasn't wasted by me to calculate it .

There are an infinite number of these 'iterations', as you've called them. However, the sum of their durations is not infinite.

Here's an analogous example. Say I walk 1 meter in 1 second, the next half meter in a half a second, the next quarter meter in a quarter second, etc. Will I ever have walked two meters?

If you understand this problem, you should understand your problem.

A visual way to see how to resolve this paradox is to take an uncolored square. Now color in half of it. Again color in half of the remaining uncolored portion, that is, color in 1/4 of the square. Keep on coloring in 1/2 of the uncolored portion. As you can see you will color in 1/2 + 1/4 + 1/8 +1/16 +... You will color in an infinite number of areas but you will never color outside the square which is finite in area. So 1/2 + 1/4 + 1/8 +... converges to a finite area or number. If I remember correctly, the paradox that you gave is by the Greek Zeno. I don't think this was resolved by the Greeks who would have to wait a few thousand years.

Nice trick! There *is* nothing wrong with this approach. It is one way to show the well known fact that Y will never catch X.

I presume you're joking.

The sum of the infinite series 10+1+1/10+1/100+... is 11.111..., or eleven and one ninth.

Let the speed of Y be 10, and the speed of X be 1, as stated above.

Let X's lead be 100.

At 111.11111..., Y catches X.

Proof:

For a body moving at constant velocity, the distance covered is equal to velocity x time.

X travels at 1, and starts at 100. To reach 111.111..., then, he must travel for 11.111... seconds.

Y travels at 10, and starts at 0. To reach 111.111..., how long will he take?

10 = 111.111... * t

t = 111.111... / 10

t = 11.111...

QED.

Another proof that Y not only catches but surpasses X, again using s = vt:

After 20 seconds, X will have travelled 20, putting him at the 120 marker. After 20 seconds, Y will have travelled 200, putting him at the 200 marker, a full 80 units ahead.

No, no, it's perfectly serious. It's been empirically tested. The Greeks knew this, it was a well-known result recorded by Aesop.