Friday, January 18, 2013

A Pursuit Curve Problem


   As a math and science educator I subscribe to a number of professional periodicals. One of  my favorites is THE PHYSICS TEACHER. I was excited to find the following problem in the September 2012 issue.

Physics Challenge for Teachers and Students
Boris Korsunsky, Column Editor
West High School, Weston, MA 02493

A Futile Chase
Two turtles, A and B are relaxing at the water’s edge a distance d apart. Then A begins to swim away from the shore. B gives chase, taking off at the same moment. During the chase A keeps swimming directly away from the shore while B keeps swimming directly towards A. The speeds of both turtles are the same. Find the distance between A and B after a long time interval.           

   As soon as I read this "challenge" problem I quickly decided not to pursue a purely mathematical solution but, instead, model the problem in Scratch. I couldn't resist the  fun of making two sprites in the form of turtles and then animating both turtles according to the conditions stated in the problem!
   Here is a diagram of the starting position for both turtles.
   The chase is futile because turtle B will never catch turtle A. Why? Both turtles travel with the same velocity, they are not traveling towards each other, and they start the chase with a separation distance d. Still, the geometry of the situation guarantees that turtle B will get closer to turtle A. What's the closest turtle B gets to turtle A? In other words, what is d after a long time interval?

   I wrote the Scratch scripts, varied the starting distance d, looked at the separation distance when both turtles were traveling in the same direction, and formed a conjecture. Doing the math to prove my conjecture seems pointless to me because I so strongly believe my conjecture to be true!
   If you would like to form your own conjecture, here's the link to my Scratch program you can download and play with.
http://scratch.mit.edu/projects/popswilson/3045644
   This is a short video of the program in action.

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