Somewhere on the high seas smuggler Stan is attempting, without much
luck, to outspeed coast guard Charlie, whose boat can go faster than Stan's. Charlie
is one mile east of Stan when a heavy fog descends. It's so heavy that
nobody can see or hear anything further than a few feet. Immediately
after the fog descends, Stan changes course and attempts to escape at
constant speed under a new, fixed course. Meanwhile, Charlie has lost track
of Stan. But Charlie happens to know Stan's speed, that it is constant, and that Stan
is sticking to some fixed heading, unknown to Charlie.
How does Charlie catch Stan?
Charlie may change course and speed at will. He knows his own speed and
course at all times. There is no wind, Charlie does not have radio or radar,
there is enough space for maneuvering, etc.
I have to admit, Bob Smith has it right, but only under certain conditions. Otherwise, the logarithmic spiral solution will not work.
If I did the math right, the most critical condition is that the maximum speed of Charlie's boat must be at least 9.12 times the speed that Stan is moving in order to guarantee an intercept. Once on the spiral path, Charlie must maintain a speed equal to 9.12 times Stan's speed in order to stay on course to intercept. (Note: If Charlie's boat is capable of a top speed of 60 knots, Stan can improve his chances of avoiding capture by travelling faster than 6.6 knots. Charlie cannot guarantee an intercept if Stan's speed is any faster than that.)
My solution requires Charlie to travel at his max speed towards Stan's last known position until he reaches a distance from that point that corresponds to the distance Stan has travelled from the same point based on his known constant speed. From here, Charlie will follow the logarithmic spiral path, always matching Stan's distance from the center.
At 60 knots, he will intercept Stan in under 2 minutes and 15 seconds.
I'm prone to making math errors, so please work it out yourself and let me know if I did it again.
Clearing away the fog: solution and admission
