On a nice summer's day a stream carries water downstream at a uniform speed of a. The banks of the stream are parallel and are separated by a distance c. Captain Jack Sparrow is on one side of the bank with his ship, the Black Pearl, perpendicular to it. Directly across from his ship, on the other bank is a pot of gold.

Jack Sparrow pulls out a compass which will always point towards the pot of gold. He keeps the the Black Pearl always headed in that direction (despite the stream pushing his ship downstream) and sets off with constant speed b.

What is the path of the boat, considering that the boat may move faster, slower, or at the same speed as the stream?

Note: The boat's path can be modeled using the Cartesian coordinate system: let the y axis be the bank with the gold and let the line x=c be the opposite bank, with the Black Pearl starting at (c,0) and the gold at (0,0).

If the coordinate system were to be stationary with respect to the water in the stream, so that the bank of the stream along with the gold is moving at speed a, one case of this problem--specifically with c= 100 and b=3a--is given in the Farmer and the Pig puzzle. A suitable transformation of coordinates is needed to account for the current problem's being relative to the gold--equivalent to using the pig's frame of reference as being stationary in the F&P problem.

Edited on December 22, 2008, 2:40 pm

Posted by Charlie on 2008-12-22 14:38:15