A rower rows regularly on a river, from A to B and back. He’s got into the habit of rowing harder when going upstream, so that he goes twice as fast relative to the water as when rowing downstream. One day as he’s rowing upstream he passes a floating bottle. He ignores it at first but then gradually grows curious about its contents. After 20 minutes of arguing with himself he stops rowing and drifts for 15 minutes. Then he sets out after the bottle. After some time rowing downstream he changes his mind, turns around, and makes his way upstream again. But his curiosity takes hold once more, and after 10 minutes of rowing upstream he turns and goes after the bottle again. Again he grows ashamed of his childishness and turns around. But after rowing upstream for 5 minutes he can’t stand it any longer, rows downstream, and picks up the bottle 1 kilometer from the point where he’d passed it. How fast is the current?
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Between passing the bottle and picking it up, the rower spends a total of 20 + 10 + 5 = 35 minutes rowing upstream. In this period his total motion relative to the bottle (and to the water) is zero — the distance he’s traveled relative to the water’s surface is the same upstream as down. We know he moves half as fast relative to the water when he’s rowing downstream as when he’s rowing upstream; this means he’s spent twice as much time rowing downstream as up, or 70 minutes. He also spent 15 minutes drifting, so altogether he picks up the bottle 35 + 70 + 15 = 120 minutes, or two hours, after passing it. The bottle has covered 1 kilometer in this time, so the current is moving at 1/2 kilometer per hour.
(A simpler way to think about the timing is to imagine that the whole thing takes place on a lake rather than a river. The rower departs from the bottle, rows 35 minutes in one direction, sits motionless for 15 minutes, and rows back to the bottle at half his earlier speed. When he reaches the bottle his round trip has taken him 35 + 15 + 70 = 120 minutes, or two hours.)
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