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diamgram of my problem

Where $\theta=45^\circ$, $d_1=200\:\mathrm{m}$, and $d_2=150\:\mathrm{m}$. I do not know how to combine the following equations: $$\begin{align} t&=\frac{d_1}{v_s\cos(45^\circ)}\\ t&=\frac{d_2}{v_r-v_s \sin(45^\circ)} \end{align}$$ Now these equation are equal to each other, but I do not know how to combine them to create an equation that solves for $v_s$. What is the process to find this equation?

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closed as off-topic by Brandon Enright, Abhimanyu Pallavi Sudhir, jinawee, John Rennie, Dilaton Mar 2 '14 at 10:10

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You've done the "most difficult" which is to get as many independant equations as needed to solve for the unknowns. The first one flows from the fact that the vertical component of the swimmer's velocity is the only one to make him reach the end of the river, and the second one from the fact that both the current and the horizontal component of the swimmer's velocity make him reach the end of the river further downstream.

You just have to consider that t is the time to reach B, which satisfies both equations: $$t=\frac{d_1}{v_s.cos(45°)}=\frac{d_2}{v_r-v_s.sin(45°)}$$ $$\Leftrightarrow d_2.v_s.cos(45°)=d_1.(v_r-v_s.sin(45°))$$ $$\Leftrightarrow v_s.(d_2.cos(45°)+d_1.(sin(45)))=d_1.v_r$$ $$\Leftrightarrow v_s=\frac{d_1.v_r}{d_2.cos(45°)+d_1.(sin(45))}$$

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  • $\begingroup$ Thank you for the detailed explanation just one question do the arrows mean same as or becomes? $\endgroup$ – user41607 Mar 2 '14 at 2:36
  • $\begingroup$ Say again? The arrows above u and v on the diagram? Vectors, look it up, it's further down the road in your studies I assume. $\endgroup$ – Mister Mystère Mar 2 '14 at 11:17
  • $\begingroup$ Oh, right. The arrows in my equations. It means "is equivalent to", it means it's exactly the same, you don't lose any information going from one to the other. It's basically two "implies" (not sure about the term in english) in both directions at the same time - implications may lose information and hence allow only making conclusions in one way (you can't go back solving for something for example). $\endgroup$ – Mister Mystère Mar 2 '14 at 15:16

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