How fast is the stream?

Thomas is going downstream on a motorboat and passes a raft at point A A . Exactly 60 60 m i n u t e s minutes after that, his motorboat turns around to go back upstream. In this return journey, he notices the same raft at a distance of 6 km \SI{6}{ \kilo\meter} from point A A .

Find the flow velocity of the stream ( ( in km/h ) \text{km/h}) assuming the duty of the engine to be constant.


The answer is 3.

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2 solutions

Assume v B , v R v_B,v_R to be the speeds of the boat and the river respectively.

Let the boat covers x x distance in 1 1 hr. with a speed v B + v R v_B+v_R since downstream ,therefore x = v B + v R x=v_B+v_R

Now when it returns at the edge of 6 6 kms before the point A he revisits the raft. Int his upstream the distance he travelled is v B + v R 6 v_B+v_R-6 with a speed v B v R v_B-v_R

So total time consumed, T = 1 + v B + v R 6 v B v R \displaystyle T = 1+\dfrac{v_B+v_R-6}{v_B-v_R}

Again this is the time in which the raft traversed 6 6 km with the flow of the river so we must have T = 6 v R T=\dfrac{6}{v_R}

1 + v B + v R 6 v B v R = 6 v R v B v R + v B + v R 6 = 6 v R ( v B v R ) v R ( 2 v B 6 ) = 6 ( v B v R ) v B v R = 3 v B v R = 3 \displaystyle \begin{aligned} 1+\dfrac{v_B+v_R-6}{v_B-v_R}=\dfrac{6}{v_R}\\ v_B-v_R+v_B+v_R-6=\dfrac{6}{v_R}\left(v_B-v_R \right) \\ v_R(2v_B-6)=6(v_B-v_R) \\ v_B v_R=3v_B \\ v_R=\boxed{3}\end{aligned}

Himanshu Thakker
Apr 25, 2017

As the boat will travel at a constant speed wrt to river therefore the relative speed is same for both upstream and downstream. Therefore it takes same time to reach the raft (60 Min or 1 Hour) as raft is floating along river. The raft has traveled a distance of 6 km wrt ground, therefore the speed is 3 kmph.

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