Question

Traveling along a river, a man in a row boat loses his hat when he passes under a bridge. In 15 minutes, he realizes that he lost his hat, and rows his boat the other way until he is able to retrieve his hat 1 km away from the bridge. What is the speed of the river's current?

Answer 1

The speed of the current is 33.33 meter/minute.

Explanation:

Let the speed of the boat in still water is x km per minute and the water is r km per minute.

The speed of the boat in downstream is (x + r) km per minute and in upstream is (x - r) km per minute.

First, the man was moving in upstream, because the hat was retrieved 1 km away from the bridge.

After losing the hat the man traveled 15 minutes upstream, hence he went 15(x - r) km far from the bridge.

After 15 minutes he was coming back to collect his hats.

Let, he has traveled downstream for t minutes to get his hat.

Hence, t(x + r) = 15r + tr + 15(x - r) or, tx = 15x or, t = 15.

Again, 15r + 15r = 1 km or, r = 1/30 km = 1000/30 =33.33 meters.

Answer 2

2 km/h

Explanation:

You want to know the speed of the river's current if a man loses his hat in the river and continues to row downstream for 15 minutes before returning upstream to retrieve his hat, 1 km downstream from where he lost it.

Reference

Since the hat is moving with the current, the boat is essentially in "still water" relative to the hat. The distance rowed in 15 minutes, relative to the hat, is exactly the distance that must be rowed to retrieve the hat. That is, the return trip takes 15 minutes as well.

The man is away from his hat for 1/2 hour, in which time the current has moved it downstream 1 km. Hence the speed of the current is ...

(1 km)/(1/2 h) = 2 km/h

The speed of the river's current is 2km/h.

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Additional comment

Interestingly, it does not matter if the original direction is upstream or downstream, nor does it matter how fast the man is rowing (as long as his speed is not zero).