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