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