Zeno's Paradoxes

The Archer's Arrow

An archer picks up an arrow, places it agains the bowstring, draws the string back and lets the arrow fly towards the target. It looks like it will be a bulls-eye, but of course we won't know for sure until it hits the target.

Before the arrow reaches the target, it must first travel half of the distance to the target. From there the arrow travels half of the remaining distance to the target. Quickly, the arrow travels half of the distance which remains after that, then half of the distance that is still between it and the target. In fact, before it can get to the target, the arrow must always first go half of the distance that remains between it and the target.

Gee.... Does the arrow ever hit the target?

The Tortoise and the Hare

The Hare was a terrible braggart. It was true that he could run very fast. In fact, it was doubted that anyone could run faster. But instead of being quietly proud of what he could do, he was always challenging the other animals to races so that he could beat them and make fun of them because they lost and he won.

One day, he challenged the Tortoise to a race. Of course she said no, because she walks so very slow. She's faster than most tortoises---she can actually go half as fast as the Hare, but that still doesn't give her any hope in a race. The Hare wanted to race with someone so bad that day, that he told her he would give her a head start that was halfway to the finish line. A head start that was halfway there. Since she could walk half as fast as the Hare could run, it might just be worth a try. She said yes.

Everyone came out to watch. The race course was set up from one end of the field to the other. The exact halfway point was paced off and marked. That is where the Tortoise started. The Hare started at the "Start" line. At the sound of the starting gun, they both took off.

The Hare ran as fast as he could and quickly reached the halfway point where the Tortoise had started. Of course the Tortoise was no longer there. She had covered half of the distance to the Finish line by then. The Hare would have to catch up to her before he could pass her. He took careful note of where she was, put his head down, and raced to that point. But when he got there, of course, the Tortoise was no longer there, because she had traveled half the distance from there to the Finish line by then. So the Hare took careful note of where she was, put his head down, and raced to where the Tortoise was. But he couldn't pass her when he got there, because by then she was gone. Once again, she had walked half the distance from where she had been to the finish line.

They did this all afternoon and on into the night. In fact, they might be doing it still. Do you think that the race ever ends? Does the Hare ever catch up to the Tortoise?