A small airplane is flying due north at $150\,\rm km/h$ when it encounters a wind of $80\,\rm km/h$ from the east. what is the resultant ground velocity of the airplane?
[Math] A small airplane is flying due north at $150\,\rm km/h$ when it encounters a wind of $80\,\rm km/h$ from the east.
calculusvectors
Best Answer
To solve this problem, you have to do a vector addition. Let $\vec v=150\,\rm km/h\,\,\text{(north)}$ and let $\vec u=80\,\rm km/h\,\,\text{(east)}$. If we project those vectors into a Cartesian coordinate system, and assuming that the unit vectors $\hat i=(1,0)$ and $\hat v=(0,1)$, we get that the the vector $\color{red}{\vec v}+\color{green}{\vec u}=\color{blue}{\vec w}$ is: $$\color{blue}{\vec w}=\color{red}{\begin{pmatrix}0\\150 \end{pmatrix}}+\color{green}{\begin{pmatrix}-80\\0 \end{pmatrix}}=\color{blue}{\pmatrix{-80\\150}}.$$ So, by the distance formula: $$\|\color{blue}{\vec w\,}\|=\sqrt{(-80)^2+(150)^2}=\sqrt{\color{white}{\overline{\color{black}{6400+22500}}}}=\sqrt{\color{white}{\overline{\color{black}{28900}}}}=170.$$
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$\color{white}{\text{this post assumes that $\gamma=0$ and so ignores the effects of special relativity and uses only the traditional velocity addition formula.}}$