Somewhere on the high sees smuggler S is attempting, without much luck, to outspeed coast guard G, whose boat can go faster than S's. G is one mile east of S 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, S changes course and attempts to escape at constant speed under a new, fixed course. Meanwhile, G has lost track of S. But G happens to know S's speed, that it is constant, and that S is sticking to some fixed heading, unknown to G.
How does G catch S?
G may change course and speed at will. He knows his own speed and course at all times. There is no wind, G does not have radio or radar, there is enough space for maneuvering, etc.