finding the theoretical speed based on current speed

Question

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?

Answer 1

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.

Answer 2

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \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}