Sunday, October 03, 2004
River-Hat Problem
You are paddling your canoe upstream at a constant velocity. After paddling for six miles, the wind blows your hat into the stream and the hat begins flowing downstream. You continue to paddle upstream for two more hours before noticing that your hat is missing, at which time you turn around and paddle downstream at the same rate you had paddled upstream, overtaking your hat just as you return to your original starting point. What is the speed of the current?
Answer:
Let c be the speed of the current, b the speed of the boat (in stillwater). The time it took for the hat to float back to its startingpoint is 6/c. The distance the boat travelled (upstream) in the two hour period after losing the hat is 2(b - c) miles, so the total upstream distance travelled is 6 + 2(b - c). During the return trip the boat was moving at a rate of (b+c) so the time from when the hat was lost until the return of the boat to the starting point was [6 + 2(b - c)]/(b + c) + 2. Setting this quantity equal to 6/c and solving gives c = 1.5, independent of b.
Aliter: Relative to the water, the hat stays still. So, again relative to the water and hat, the rower moves just as fast upstream as downstream. After the hat dropped, the rower rowed for four hours and the hat moved six miles, so the current is flowing at a rate of 6/4 = 1.5 miles per hour.
Answer:
Let c be the speed of the current, b the speed of the boat (in stillwater). The time it took for the hat to float back to its startingpoint is 6/c. The distance the boat travelled (upstream) in the two hour period after losing the hat is 2(b - c) miles, so the total upstream distance travelled is 6 + 2(b - c). During the return trip the boat was moving at a rate of (b+c) so the time from when the hat was lost until the return of the boat to the starting point was [6 + 2(b - c)]/(b + c) + 2. Setting this quantity equal to 6/c and solving gives c = 1.5, independent of b.
Aliter: Relative to the water, the hat stays still. So, again relative to the water and hat, the rower moves just as fast upstream as downstream. After the hat dropped, the rower rowed for four hours and the hat moved six miles, so the current is flowing at a rate of 6/4 = 1.5 miles per hour.
