Head Start Hare

This isn't really a puzzle... But it makes for an interesting thought exercise for those who look at the trees rather than the forest.

A turtle and a hare decide to have a race. The turtle has finally figured out a way, he thinks, that will guarantee him a win, or at least not a loss. And, if he has figured it correctly, it doesn't matter how fast the hare can run! The only stipulation is that he has to have a head start... Here's his thinking:

Say the turtle gets an X minute head start, and travels at Y miles/hour. Then the hare takes off at Z miles/hour (Z >> Y). Here's the idea... After a few moments, the turtle gets to point "a". The hare, of course, takes some amount of time to catch up to point "a". But by then, the turtle has gone a bit further, to point "b". The hare will take some finite amount of additional time to get to point "b". By then, the turtle will have moved on to point "c". Etc, etc, etc....  It appears, to the turtle, that the hare will never catch up!

Clearly, the turtle is a little "slow". We know the hare will win give reasonable values for speed, race distance, and the head start. Using simple math, we can figure out the point that the hare overtakes the turtle. So, where did the turtle go wrong in his thinking?



Comments:

Because the time and distance intervals are dropping each instance, the turtle is really just measuring the time until he's caught, just before which event the distance and time slices become infinitesimal. In real life, with the same time slice for each unit of distance travelled, or the same distance measured per time to travel, he has simply extended the amount of time he has before he's passed.

Posted by David Pipes on March 07, 2005 at 05:27 AM EST #

You wouldn't perchance have been recently reading <a href=http://www.amazon.com/exec/obidos/ASIN/0465026567/102-7809004-7876941"Gödel, Escher, Bach: An Eternal Golden Braid recently would you?

As I recall this is one of the things discussed early on.

Alan.

Posted by Alan Hargreaves on March 07, 2005 at 09:13 AM EST #

This is one of the fundamental concepts of infinite series. I think it is and should be taught in all sophomore undergrad college curriculum.

Posted by whatsinaname on March 10, 2005 at 02:24 AM EST #

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