Problem of the Week: Chase!
Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s). Each week’s entries are scored by a panel of faculty judges, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button. The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.
The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited! Send solutions to firstname.lastname@example.org or put solutions in the marked envelope in the hallway outside Glatfelter 215. This problem was posted on Friday, January 25 and solutions are due on Friday, February 1 by 5:00 p.m.
A rabbit is going to come out of its hole (located at point (0,0) in the plane) at time t = 0 and run in a straight line, at speed 1, to another hole located at point (1, 0) in the plane.
At time t = 0, a dog is standing at point (p, 1) in the plane, where 0 ≤ p ≤ 1. While the rabbit runs, the dog chases it. The dog runs at constant speed r > 0 and always runs directly toward the rabbit.
QUESTION: Say as much as you can about what values of p and r allow the dog to catch the rabbit. Solutions that include computer-aided investigations are encouraged!