finding the theoretical speed based on current speed A bird is attempting to fly northeast at a constant speed, but a wind blowing southward at 5 miles per hour blows the bird off course. If the bird’s overall movement (incorporating its intended movement and the movement due to wind) is at a $\sqrt{53}$ miles per hour, how fast would it have been traveling if there was no wind?
 A: Suppose the bird's intended movement is $x$ mph north and $x$ mph east at once, so $\sqrt2x$ mph northeast in total. The wind means that the bird is actually travelling $x-5$ mph north, so by the Pythagorean theorem we have $x^2+(x-5)^2=53$ or $x=7$. So the bird's speed without wind is $7\sqrt2$ mph.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
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\begin{align}
&\vec{v}_{bird} = v_{bird}\cos\pars{\pi \over 4}\hat{x}+
v_{bird}\sin\pars{\pi \over 4}\hat{y}\quad\pars{~\color{red}{without}\
\mbox{ wind}~}
\\[5mm]
&\vec{v}_{bird+} = v_{bird}\cos\pars{\pi \over 4}\hat{x} +
\bracks{v_{bird}\sin\pars{\pi \over 4} - 5}\hat{y}\quad\pars{~\color{red}{with}\
\mbox{wind}~}
\\[5mm]
& 53 = v_{bird+} =
\left.\root{{v_{bird}^{2} \over 2} + \pars{{v_{bird} \over \root{2}} - 5}^{2}}
\right\vert_{\large\ v_{bird}\ \geq\ 0}
&\\[5mm] \implies &
\bbx{v_{bird} = 7\root{2}\ {\mrm{m} \over \mrm{h}}
\approx 9.8995\ {\mrm{m} \over \mrm{h}}}\\ &
\end{align}
