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https://www.youtube.com/watch?v=5F1XcTjpJs4 - Derivation of Gravitational Potential Energy by Rhett Allain
Eq. \ref{eq:8462685} is an initial equation. starting velocity at infinity is zero \begin{equation} v(r=\infty) = 0 \label{eq:8462685} \end{equation} Eq. \ref{eq:3470082} is an initial equation. https://en.wikipedia.org/wiki/Newton\\%27s\_law\_of\_universal\_gravitation\#Modern\_form \begin{equation} \vec{F} = G \frac{m_1 m_2}{x^2} \hat{x} \label{eq:3470082} \end{equation} Eq. \ref{eq:9835406} is an initial equation. \begin{equation} W_{\rm by\ system} = \Delta KE \label{eq:9835406} \end{equation} Eq. \ref{eq:6798426} is an initial equation. \begin{equation} W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r} \label{eq:6798426} \end{equation} Substitute LHS of Eq. \ref{eq:3470082} into Eq. \ref{eq:6798426}; yields Eq. \ref{eq:7300369}. \begin{equation} W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx \label{eq:7300369} \end{equation} Simplify Eq. \ref{eq:7300369}; yields Eq. \ref{eq:9707318}. \begin{equation} W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx \label{eq:9707318} \end{equation} Evaluate definite integral Eq. \ref{eq:9707318}; yields Eq. \ref{eq:5818573}. \begin{equation} W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right) \label{eq:5818573} \end{equation} Simplify Eq. \ref{eq:5818573}; yields Eq. \ref{eq:2429271}. \begin{equation} W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right) \label{eq:2429271} \end{equation} Simplify Eq. \ref{eq:2429271}; yields Eq. \ref{eq:4947999}. \begin{equation} W_{\rm to\ system} = \frac{G m_1 m_2}{r} \label{eq:4947999} \end{equation} Eq. \ref{eq:9781919} is an initial equation. \begin{equation} \Delta KE = KE_{\rm final} - KE_{\rm initial} \label{eq:9781919} \end{equation} Eq. \ref{eq:5104592} is an initial equation. \begin{equation} KE = \frac{1}{2} m v^2 \label{eq:5104592} \end{equation} Substitute LHS of Eq. \ref{eq:9781919} into Eq. \ref{eq:9835406}; yields Eq. \ref{eq:8118190}. \begin{equation} W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial} \label{eq:8118190} \end{equation} Change of variable \(KE\) to \(KE_{\rm initial}\) and \(m\) to \(m_1\) and \(v\) to \(v_{\rm initial}\) in Eq. \ref{eq:5104592}; yields Eq. \ref{eq:9031887}. \begin{equation} KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2 \label{eq:9031887} \end{equation} Change of variable \(KE\) to \(KE_{\rm final}\) and \(m\) to \(m_1\) and \(v\) to \(v_{\rm final}\) in Eq. \ref{eq:5104592}; yields Eq. \ref{eq:1397156}. \begin{equation} KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2 \label{eq:1397156} \end{equation} Eq. \ref{eq:1712972} is an initial equation. \begin{equation} v_{\rm initial} = v(r=\infty) \label{eq:1712972} \end{equation} Substitute LHS of Eq. \ref{eq:8462685} into Eq. \ref{eq:1712972}; yields Eq. \ref{eq:6923850}. \begin{equation} v_{\rm initial} = 0 \label{eq:6923850} \end{equation} Substitute LHS of Eq. \ref{eq:6923850} into Eq. \ref{eq:9031887}; yields Eq. \ref{eq:7110498}. \begin{equation} KE_{\rm initial} = 0 \label{eq:7110498} \end{equation} Substitute LHS of Eq. \ref{eq:7110498} into Eq. \ref{eq:8118190}; yields Eq. \ref{eq:2751634}. \begin{equation} W_{\rm by\ system} = KE_{\rm final} \label{eq:2751634} \end{equation} Substitute LHS of Eq. \ref{eq:1397156} into Eq. \ref{eq:2751634}; yields Eq. \ref{eq:6536576}. \begin{equation} W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2 \label{eq:6536576} \end{equation} Eq. \ref{eq:2619766} is an initial equation. \begin{equation} W_{\rm by\ system} = W_{\rm to\ system} \label{eq:2619766} \end{equation} Substitute LHS of Eq. \ref{eq:6536576} into Eq. \ref{eq:2619766}; yields Eq. \ref{eq:8655239}. \begin{equation} \frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system} \label{eq:8655239} \end{equation} Substitute LHS of Eq. \ref{eq:4947999} into Eq. \ref{eq:8655239}; yields Eq. \ref{eq:2942416}. \begin{equation} \frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r} \label{eq:2942416} \end{equation} Multiply both sides of Eq. \ref{eq:2942416} by \(2/m_1\); yields Eq. \ref{eq:4594601}. \begin{equation} v_{\rm final}^2 = \frac{2 G m_2}{r} \label{eq:4594601} \end{equation} Take the square root of both sides of Eq. \ref{eq:4594601}; yields Eq. \ref{eq:7112224} and Eq. \ref{eq:1366396}. \begin{equation} v_{\rm final} = \sqrt{\frac{2 G m_2}{r}} \label{eq:7112224} \end{equation} \begin{equation} v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}} \label{eq:1366396} \end{equation} Change variable \(v_{\rm final}\) to \(v(r)\) in Eq. \ref{eq:7112224}; yields Eq. \ref{eq:3435796}. \begin{equation} v(r) = \sqrt{\frac{2 G m_2}{r}} \label{eq:3435796} \end{equation} Eq. \ref{eq:3435796} is one of the final equations.