Catching a submarine of unknown heading

Question

At time zero a submarine is at a known point P, a distance D from your aircraft. It moves at constant speed v in a straight line, but in a direction you do not know and cannot observe. Your aircraft flies at speed 2v and catches the submarine the instant it is directly above it. You get no further information.

Devise a flight path that is guaranteed to pass over the submarine, however it is heading.

Accepted Answer

Fly straight towards P until you are on the circle of possible submarine positions, then fly a spiral that keeps you exactly on that expanding circle while your bearing around P makes a full turn. Phase 1: reach the circle. The submarine's possible positions at time t form a circle of radius vt around P. Flying straight at P from distance D at speed 2v, your distance from P is D minus 2vt, which equals vt when t1 = D/(3v). At that moment you are on the circle, at some bearing from P. Phase 2: sweep the circle. From now on, keep your distance from P equal to vt (growing at exactly v) while als…

Catching a submarine of unknown heading

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