This looks like a variant of the "Lion and Man" problem which is also described as a "bounded pursuit" problem in the operations research journals:

http://www.google.com/search?q=lion+and+man+problem+&ie=...

-and-

http://mathworld.wolfram.com/PursuitCurve.html

-and-

http://books.google.com/books?id=HeESjfM2geUC (good!)

This problem may have had relevance in World War II all the way to the present: dogfights between airplanes, battleships trying to sink each other, etc.