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.

(In reply to new twist by Mindrod)

This is true only if he wanted to keep the same angular rate around the spiral, which isn't necessary.  It's best for Charlie to always move at his max velocity, Vc.

Best plan is for Charlie to close on Stan's last known point until at the critical distance: r1 = (Vs / (Vc + Vs) ), t1 = 1 / (Vc + Vs)

At this point he begins his spiral.  If I did the math correctly

Theta = (((Vc/Vs)^2-1)^1/2) log(t2/t1)

where t2 is the time at angle theta.  If Charlie has to go full circle, then theta is 2pi and we can solve for velocity ratios.

Vc/Vs = 2, t1 = .33, t2 = 1419.1

Vc/Vs = 3, t1 = .25, t2 = 41.6

Total time is t1 + t2.

I have a nagging suspicion that I missed something.  Ah well.

Posted by Bob Smith on 2005-11-02 12:08:17