[Physics] Finding an equation to solve for the velocity of a swimmer crossing a river

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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?

Best Answer

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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