<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\begin{document}
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{\bf Question}

Show that the speed of a planet in a planetary orbit satisfies
$$v^2 = \frac{\mu}{m}\left\{\frac{2}{r} - \frac{1}{a} \right\}.$$

\vspace{.25in}

{\bf Answer}

Use the energy equation $\ds \frac{1}{2}mv^2 - \frac{\mu}{r} = E$

But $\ds a = -\frac{\mu}{2E}$

Therefore $\ds \frac{1}{2}mv^2 - \frac{\mu}{r} = -\frac{\mu}{2a}
\Rightarrow v^2 = \frac{\mu}{m} \left( \frac{2}{r} - \frac{1}{a}
\right)$



\end{document}
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