[Phone Booth]

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We need to figure out the speed that the paddler needs to travel to go 1 hour north and then turn around and cover his previous distance + 2 miles in the time that the stream carries the log 2 miles. You figure that if the canoeist is using the same force in both directions, his speed would obviously differ depending on which way he was travelling. If we assume he'd travel a certain speed in still water, he'd theoretically travel that speed - the stream's speed when heading upstream, and that speed + the stream's speed heading downstream.

So if the stream's speed is 1 mph - the paddler's speed would need to be 3 mph...he would go 2 miles upstream in the first hour(3mph - 1mph), and then in the next hour he would come back 4 miles to catch the log - having traveled 2 miles upstream and 4 miles downstream over the course of these 2 hours. The log would travel the 2 miles over the course of 2 hours(at 1mph)

James

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I don't follow - as far as I can see, the paddler's speed is an indeterminate (assuming that the problem doesn't specify how long it took him to go the first three miles - the wording is unclear - if it does, it's a bad question).

Let paddler's speed be p and stream speed be s. After 2 miles *and* one hour, paddler's position is at 2 + p - s from the starting position. The log is at 2 - s from the paddler's starting position. The paddler's going at the speed of p + s and the log at the speed of s. The ratio of their speeds must be equal to the ratio of their distance away from the starting position.

(2 + p - s) / (2 - s) = (p + s) / s

which is same as

(p + (2 - s)) / (2 - s) = (p + s) / s

which, assuming p > 0 and s > 0, implies 2 - s = s, thus s = 1, leaving p an indeterminate.

Another way to look at it is that, if s = 1, the log takes two hours to cover two miles. It doesn't matter what the paddler's speed is, if he goes upstream for 1 hour and downstream for one hour, he will end up where the stream takes him, because he's cancelled out his own paddling.