If a man can row at a speed of x m/sec , and the water in the river flows at a speed of y m/sec , then if then man wants to cross the river from one bank to the opposite bank, i.e. he would be moving horizontally in the river , neither upstream nor downstream , then what would be the effect on the speed of rowing of the man . What would be his speed in terms of x and y ?
Effect on boat speed on Crossing a river with a boat from one bank to the opposite bank
1 Answers
Sorry for lack of picture! You will have to supply it, what is written below has no meaning without a picture. Let $A$ be the point on the bank where the person starts. The person rows at a certain angle $\theta$ from the direct path, towards the upstream side. In unit time, without the current, she would go a distance $x$. But she is swept downstream a distance $y$.
If there was no current, then rowing at speed $x$ at angle $\theta$, she would reach a certain point $B$ in the stream. But in the meantime she is swept downstream to $C$. Note that $C$ is on the line from $A$ directly towards the opposite bank. The triangle $ABC$ is right-angled at $C$. We have that the hypotenuse $AB$ is $x$, and $BC$ is $y$. So, by the Pythagorean Theorem, the distance $AC$ across the stream that she reaches is $\sqrt{x^2-y^2}$. That is the effective crossing speed.
If the above is too abstract, let $C$ be the point on the opposite bank across from the start $A$, and let $B$ be the point on the opposite bank that she would reach if she rowed at angle $\theta$ and there were no current. Let the time taken be $t$. Then $AB=xt$, $BC=yt$, so $AC=\sqrt{(xt)^2+(yt)^2}=t\sqrt{x^2-y^2}$. It follows that the effective crossing speed is $t\sqrt{x^2-y^2}/t=\sqrt{x^2-y^2}$.
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1@sqlchild: That is what my solution assumed. And it can only be done if $x\gt y$. If $x=10$ and $y=13$, whatever the person does, the person will end up on the opposite bank downstream from the hotel, and will need to walk to the hotel. – 2012-05-04