Minimizing Total Time

In a famous math problem, explored in "real life" by Tim Pennings, http://www.maa.org/features/elvisdog.pdf, a dog is to run along a straight shore, then jump into the water and start swimming to get to a ball, thrown into the water at some distance down the shore. The goal is to find the point where the dog should leave the shore, in order to minimize the time it takes to get to the ball.

In the applet, we first adjust the dog's running speed and swimming speed (in meters per second) -- the dog in the experiment was a Corgi, so the speeds are not super-fast. Then we "throw" the ball, and finally we slide the jump-off point around to try to find the minimal time.

1. "Fix" the speeds and the position of the ball. Then Slide the jump-off point from left to right, and observe the total times. By plotting some points, sketch out the graph of the Time function, in terms of the parameter, J = jump-off point.

2. Repeat #1 for different ratios of r to s. As Dr. Pennings does in his paper, explore the ratio of y/x for fixed values of r and s, and different distances, x. See if you can experimentally arrive at his optimal ratio of y = 0.144x for his reported values of r and s.

Dave Matthews, Created with GeoGebra