% this tex file was generated by the Physics Derivation Graph 
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage[dvipdfmx,colorlinks=true,pdfkeywords={physics derivation graph}]{hyperref}
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% inference rules as newcommand for use in the body
\newcommand\declareinitialexpression[1]{Eq.~\ref{eq:#1} is an initial equation.}
\newcommand\substituteLHSofexprintoexpr[3]{Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\integrate[2]{Integrate Eq.~ref{eq:#1}; yields Eq.~ref{eq:#2}.}
\newcommand\evaluatedefiniteintegral[2]{Evaluate definite integral Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\changetwovariablesinexpression[6]{Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.}
\newcommand\simplify[2]{Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\substituteLHSoftwoexpressionsintoexpression[4]{Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.}
\newcommand\declareassumption[1]{Eq.~\ref{eq:#1} is an assumption.}
\newcommand\addXtobothsides[3]{Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\multiplybothsidesby[3]{Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.}
\newcommand\swapLHSwithRHS[2]{Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.}
\newcommand\squarerootbothsides[3]{Take the square root of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2} and Eq.~\ref{eq:#3}.}
\newcommand\declarefinalexpression[1]{Eq.~\ref{eq:#1} is one of the final equations.}
\title{escape velocity}
\date{\today}
%\author{a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}
\begin{document}
\maketitle
\begin{abstract}
Generated by the \href{http://allofphysics.com/}{Physics Derivation Graph}.
\end{abstract}

% step ID = 0006881790
\declareinitialexpression{069718f90e26dd96d1bf009b6e25ba70eb19452bb2395613dee733313515cae7}
\begin{equation}
F = G \frac{m_1 m_2}{x^2}
\label{eq:069718f90e26dd96d1bf009b6e25ba70eb19452bb2395613dee733313515cae7}
\end{equation}

% step ID = 0004650277
\substituteLHSofexprintoexpr{861dc4e3626021e2be9a34cb73d0003e145e0542b8a784e129d71d63f8b93075}{069718f90e26dd96d1bf009b6e25ba70eb19452bb2395613dee733313515cae7}{2d5a719e01b4155d8138d87a649b065d983c98295787b71f5d6f711d941ced2e}
\begin{equation}
dW = G \frac{m_1 m_2}{x^2} dx
\label{eq:2d5a719e01b4155d8138d87a649b065d983c98295787b71f5d6f711d941ced2e}
\end{equation}

% step ID = 0006008233
\integrate{2d5a719e01b4155d8138d87a649b065d983c98295787b71f5d6f711d941ced2e}{b0c3dff2411522dbfe1bf7515dba8864f026f201bac3dd97e6a050e6aae600ee}
\begin{equation}
\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx
\label{eq:b0c3dff2411522dbfe1bf7515dba8864f026f201bac3dd97e6a050e6aae600ee}
\end{equation}

% step ID = 0003324084
\evaluatedefiniteintegral{b0c3dff2411522dbfe1bf7515dba8864f026f201bac3dd97e6a050e6aae600ee}{793f33a60d8cacbc7ef9059a8cffb8cd28692f443a230b1a452bfa52cb987f80}
\begin{equation}
W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)
\label{eq:793f33a60d8cacbc7ef9059a8cffb8cd28692f443a230b1a452bfa52cb987f80}
\end{equation}

% step ID = 0008039319
\changetwovariablesinexpression{m}{m_2}{m_{\rm Earth}}{m_1}{793f33a60d8cacbc7ef9059a8cffb8cd28692f443a230b1a452bfa52cb987f80}{c0c8209e410b2bd448eccb44247f5a98222a64d912de1690f8842fb4be0b0534}
\begin{equation}
W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)
\label{eq:c0c8209e410b2bd448eccb44247f5a98222a64d912de1690f8842fb4be0b0534}
\end{equation}

% step ID = 0009534004
\simplify{c0c8209e410b2bd448eccb44247f5a98222a64d912de1690f8842fb4be0b0534}{6aec9d31cbac6eca870979e15fa2985dde04595493ad369e3a492ac649a606e4}
\begin{equation}
W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)
\label{eq:6aec9d31cbac6eca870979e15fa2985dde04595493ad369e3a492ac649a606e4}
\end{equation}

% step ID = 0004165232
\simplify{6aec9d31cbac6eca870979e15fa2985dde04595493ad369e3a492ac649a606e4}{28b575d77192bc712410335bfa9c6f3f792ecd9752fa766e89ef51826f6ed090}
\begin{equation}
W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}
\label{eq:28b575d77192bc712410335bfa9c6f3f792ecd9752fa766e89ef51826f6ed090}
\end{equation}

% step ID = 0001179540
\substituteLHSoftwoexpressionsintoexpression{a2d612ff20da41ba51fb289d197daec1ee3f6603114e67e4cab66bba19549f95}{64698b1761ff8fec62b8bb445ae3ae7f5d3cb36b34b961374fda5e1f962023a7}{89d2b509e30086e719745e106373741089371676ae2a3327418be5e1e8a017fc}{d2e9748f74c4cfca47ddaee3157b80fa2d700f753c6bc1811b53ad632098c54b}
\begin{equation}
KE_2 + PE_2 = KE_1 + PE_1
\label{eq:d2e9748f74c4cfca47ddaee3157b80fa2d700f753c6bc1811b53ad632098c54b}
\end{equation}

% step ID = 0001518987
\declareassumption{87d0d78a400c9599bd98ba8c009d50f51c079f349af2bed9a66ad0c6a1e0f79e}
\begin{equation}
PE_2 = 0
\label{eq:87d0d78a400c9599bd98ba8c009d50f51c079f349af2bed9a66ad0c6a1e0f79e}
\end{equation}

% step ID = 0006988834
\declareassumption{369b7951a9d3491a32ea6b81014a5f051b71ef1e4ac787b262cd6d4f7befb08c}
\begin{equation}
KE_2 = 0
\label{eq:369b7951a9d3491a32ea6b81014a5f051b71ef1e4ac787b262cd6d4f7befb08c}
\end{equation}

% step ID = 0007320847
\substituteLHSoftwoexpressionsintoexpression{d2e9748f74c4cfca47ddaee3157b80fa2d700f753c6bc1811b53ad632098c54b}{369b7951a9d3491a32ea6b81014a5f051b71ef1e4ac787b262cd6d4f7befb08c}{87d0d78a400c9599bd98ba8c009d50f51c079f349af2bed9a66ad0c6a1e0f79e}{e030d39c146f05dd906a244c8ad452a1f41b8a5b2d29f102f0497c7859231a4d}
\begin{equation}
0 = KE_1 + PE_1
\label{eq:e030d39c146f05dd906a244c8ad452a1f41b8a5b2d29f102f0497c7859231a4d}
\end{equation}

% step ID = 0009709070
\changetwovariablesinexpression{PE_{\rm Earth\ surface}}{PE_1}{KE_{\rm escape}}{KE_1}{e030d39c146f05dd906a244c8ad452a1f41b8a5b2d29f102f0497c7859231a4d}{168121234b5f2d834a4e236bdeea0ae0ee91fd6e240a634410443a350e739a8e}
\begin{equation}
0 = KE_{\rm escape} + PE_{\rm Earth\ surface}
\label{eq:168121234b5f2d834a4e236bdeea0ae0ee91fd6e240a634410443a350e739a8e}
\end{equation}

% step ID = 0001868693
\declareinitialexpression{c402dcf1d3fd5fde33fbce6ff9aa0f2969314eec7a5bd83ec1f22413bc3fdd8c}
\begin{equation}
PE_{\rm Earth\ surface} = -W
\label{eq:c402dcf1d3fd5fde33fbce6ff9aa0f2969314eec7a5bd83ec1f22413bc3fdd8c}
\end{equation}

% step ID = 0006200985
\substituteLHSofexprintoexpr{c402dcf1d3fd5fde33fbce6ff9aa0f2969314eec7a5bd83ec1f22413bc3fdd8c}{28b575d77192bc712410335bfa9c6f3f792ecd9752fa766e89ef51826f6ed090}{18e2950f2ee13d2becafa4eff3679e01203400a22a6ceec99ce24932fa28892c}
\begin{equation}
PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}
\label{eq:18e2950f2ee13d2becafa4eff3679e01203400a22a6ceec99ce24932fa28892c}
\end{equation}

% step ID = 0008498604
\changetwovariablesinexpression{v_{\rm escape}}{v}{KE_{\rm escape}}{KE}{1f684c95d12c59e7f76d19968932cacfd618964d28dc4318c8194ee76f66820d}{215b75b16d0253c55a353b42bdc78c85187e19701ad86571be173364e4227c7a}
\begin{equation}
KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2
\label{eq:215b75b16d0253c55a353b42bdc78c85187e19701ad86571be173364e4227c7a}
\end{equation}

% step ID = 0008184884
\substituteLHSoftwoexpressionsintoexpression{168121234b5f2d834a4e236bdeea0ae0ee91fd6e240a634410443a350e739a8e}{18e2950f2ee13d2becafa4eff3679e01203400a22a6ceec99ce24932fa28892c}{215b75b16d0253c55a353b42bdc78c85187e19701ad86571be173364e4227c7a}{2b9452cd2b39fc40c063ddcab00ccfe8d4625d53de05ae158437c94a1944a76a}
\begin{equation}
0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2
\label{eq:2b9452cd2b39fc40c063ddcab00ccfe8d4625d53de05ae158437c94a1944a76a}
\end{equation}

% step ID = 0008921345
\addXtobothsides{G \frac{m_{\rm Earth} m}{r_{\rm Earth}}}{2b9452cd2b39fc40c063ddcab00ccfe8d4625d53de05ae158437c94a1944a76a}{0312bfe76cb17f506723eef66faef81fd795c54a8e52773adf71131da2fec9f9}
\begin{equation}
G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2
\label{eq:0312bfe76cb17f506723eef66faef81fd795c54a8e52773adf71131da2fec9f9}
\end{equation}

% step ID = 0005854765
\simplify{0312bfe76cb17f506723eef66faef81fd795c54a8e52773adf71131da2fec9f9}{17435c6e96c15e47b139bd74db4a604dfb6b7f0068d4e69a604f629179b13afa}
\begin{equation}
G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2
\label{eq:17435c6e96c15e47b139bd74db4a604dfb6b7f0068d4e69a604f629179b13afa}
\end{equation}

% step ID = 0003270895
\multiplybothsidesby{2}{17435c6e96c15e47b139bd74db4a604dfb6b7f0068d4e69a604f629179b13afa}{83b52ab7c84a6d291da909b841bc118b415965d2d7eddc8e56f2938f6bf952d9}
\begin{equation}
2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2
\label{eq:83b52ab7c84a6d291da909b841bc118b415965d2d7eddc8e56f2938f6bf952d9}
\end{equation}

% step ID = 0001078239
\swapLHSwithRHS{83b52ab7c84a6d291da909b841bc118b415965d2d7eddc8e56f2938f6bf952d9}{f534ceb1ec46c33945b21f895a80cf09c1f62694925a73623df9d0ecae170e8e}
\begin{equation}
v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}
\label{eq:f534ceb1ec46c33945b21f895a80cf09c1f62694925a73623df9d0ecae170e8e}
\end{equation}

% step ID = 0004505146
\squarerootbothsides{f534ceb1ec46c33945b21f895a80cf09c1f62694925a73623df9d0ecae170e8e}{6ed55dcccb7478dd08e496a05091a129607345d77c3daf6365e23277f8cef4c7}{f254219da36964b0014acea36ff07dbe135ad061657b92ec9b06521ff37f6cb8}
\begin{equation}
v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}
\label{eq:6ed55dcccb7478dd08e496a05091a129607345d77c3daf6365e23277f8cef4c7}
\end{equation}
\begin{equation}
v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}
\label{eq:f254219da36964b0014acea36ff07dbe135ad061657b92ec9b06521ff37f6cb8}
\end{equation}

% step ID = 0004334762
\declarefinalexpression{f254219da36964b0014acea36ff07dbe135ad061657b92ec9b06521ff37f6cb8}

replaced Earth-specific variables
% step ID = 0004437085
\changetwovariablesinexpression{r}{r_{\rm Earth}}{m}{m_{\rm Earth}}{f254219da36964b0014acea36ff07dbe135ad061657b92ec9b06521ff37f6cb8}{aa688a6d41e241a4740d31245b27122f4a29b03d27c414884ef13391b852745b}
\begin{equation}
v_{\rm escape} = \sqrt{2 G \frac{m}{r}}
\label{eq:aa688a6d41e241a4740d31245b27122f4a29b03d27c414884ef13391b852745b}
\end{equation}
\bibliographystyle{plain}
\bibliography{pdg_derivation_citations.bib}
\end{document}
% EOF
