Two students are canoeing on a river. While heading upstream, they accidentally

Question

Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for 1.3 h , reaching a point 1.3 km farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) 4.5 km downstream from the turn-around point. Part a) Assuming a constant paddling effort throughout, how fast is the river flowing? Part b) What would the canoe speed in a still lake be for the same paddling effort?

Explanation / Answer

Let the speed of the stream be v and the boat be u

Now the relative velocity of the boat in the upstream

Vup = u-v = 1.3 km / 1.3 hours

u-v = 1kmph

u – v =1                                ….(1)

relative velocity of the boat in the downstream

Vdown = u + v = 4.5 km/t hours

u+v = 4.5/t                          ….(2)

The distance bottle travelled

D = 4.5 – 1.3 = 3.2

This bottle will be travelling with the velocity of stream (v) until the student’s get it. It is moving for a period of ( 1.3 hour + t hour)

v = 3.2 / 1.3+t                        ….(3)

From (1) and (2)

1 + 2v = 4.5/t                      ….(4)

From (3) and (4)

1 + (3.2/(1.3+t)) = 4.5/t

(1.3+ t + 3.2)/(1.3+t) = 4.5/t

(1.3+ t + 3.2)t = 4.5(1.3+t)

1.3t + t2 + 3.2 t = 5.85+ 4.5t

t2 = 5.85

t = 2.42 hours

v = 3.2 / 1.3+t      

v = 3.2 / 1.3+2.42                                        

v = 0.8602 kmph

u = 1+v = 1.8602 kmph

River flowing at 0.8602 kmph speed and the canoe is moving at 1.8602 kmph speed

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