Generated by the Physics Derivation Graph. Eq. \ref{eq:3279838} is an initial equation. \begin{equation} F = G \frac{m_1 m_2}{x^2} \label{eq:3279838} \end{equation} Substitute LHS of Eq. \ref{eq:3279838} into Eq. \ref{eq:2123766}; yields Eq. \ref{eq:3686928}. \begin{equation} dW = G \frac{m_1 m_2}{x^2} dx \label{eq:3686928} \end{equation} Integrate Eq.~ref{eq:3686928}; yields Eq.~ref{eq:4803506}. \begin{equation} \int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx \label{eq:4803506} \end{equation} Evaluate definite integral Eq. \ref{eq:4803506}; yields Eq. \ref{eq:1089445}. \begin{equation} W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) \label{eq:1089445} \end{equation} Change variable \(m_1\) to \(m_{\rm Earth}\) and \(m_2\) to \(m\) in Eq. \ref{eq:1089445}; yields Eq. \ref{eq:2341415}. \begin{equation} W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right) \label{eq:2341415} \end{equation} Simplify Eq. \ref{eq:2341415}; yields Eq. \ref{eq:2190752}. \begin{equation} W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right) \label{eq:2190752} \end{equation} Simplify Eq. \ref{eq:2190752}; yields Eq. \ref{eq:2238158}. \begin{equation} W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}} \label{eq:2238158} \end{equation} Substitute LHS of Eq. \ref{eq:6310702} and LHS of Eq. \ref{eq:5160388} into Eq. \ref{eq:6330719}; yields Eq. \ref{eq:4840471}. \begin{equation} KE_2 + PE_2 = KE_1 + PE_1 \label{eq:4840471} \end{equation} Eq. \ref{eq:7682341} is an assumption. \begin{equation} PE_2 = 0 \label{eq:7682341} \end{equation} Eq. \ref{eq:9324316} is an assumption. \begin{equation} KE_2 = 0 \label{eq:9324316} \end{equation} Substitute LHS of Eq. \ref{eq:7682341} and LHS of Eq. \ref{eq:9324316} into Eq. \ref{eq:4840471}; yields Eq. \ref{eq:8369684}. \begin{equation} 0 = KE_1 + PE_1 \label{eq:8369684} \end{equation} Change variable \(KE_1\) to \(KE_{\rm escape}\) and \(PE_1\) to \(PE_{\rm Earth\ surface}\) in Eq. \ref{eq:8369684}; yields Eq. \ref{eq:9967559}. \begin{equation} 0 = KE_{\rm escape} + PE_{\rm Earth\ surface} \label{eq:9967559} \end{equation} Eq. \ref{eq:6773616} is an initial equation. \begin{equation} PE_{\rm Earth\ surface} = -W \label{eq:6773616} \end{equation} Substitute LHS of Eq. \ref{eq:2238158} into Eq. \ref{eq:6773616}; yields Eq. \ref{eq:9437784}. \begin{equation} PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} \label{eq:9437784} \end{equation} Change variable \(KE\) to \(KE_{\rm escape}\) and \(v\) to \(v_{\rm escape}\) in Eq. \ref{eq:3778087}; yields Eq. \ref{eq:5106827}. \begin{equation} KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2 \label{eq:5106827} \end{equation} Substitute LHS of Eq. \ref{eq:5106827} and LHS of Eq. \ref{eq:9437784} into Eq. \ref{eq:9967559}; yields Eq. \ref{eq:3493665}. \begin{equation} 0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2 \label{eq:3493665} \end{equation} Add \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}}\) to both sides of Eq. \ref{eq:3493665}; yields Eq. \ref{eq:6523887}. \begin{equation} G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2 \label{eq:6523887} \end{equation} Simplify Eq. \ref{eq:6523887}; yields Eq. \ref{eq:7567097}. \begin{equation} G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2 \label{eq:7567097} \end{equation} Multiply both sides of Eq. \ref{eq:7567097} by \(2\); yields Eq. \ref{eq:3358651}. \begin{equation} 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2 \label{eq:3358651} \end{equation} Swap LHS of Eq. \ref{eq:3358651} with RHS; yields Eq. \ref{eq:3908344}. \begin{equation} v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} \label{eq:3908344} \end{equation} Take the square root of both sides of Eq. \ref{eq:3908344}; yields Eq. \ref{eq:6389964} and Eq. \ref{eq:8779043}. \begin{equation} v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} \label{eq:6389964} \end{equation} \begin{equation} v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}} \label{eq:8779043} \end{equation} Eq. \ref{eq:6389964} is one of the final equations. Change variable \(m_{\rm Earth}\) to \(m\) and \(r_{\rm Earth}\) to \(r\) in Eq. \ref{eq:6389964}; yields Eq. \ref{eq:1619188}. replaced Earth-specific variables \begin{equation} v_{\rm escape} = \sqrt{2 G \frac{m}{r}} \label{eq:1619188} \end{equation}