Generated by the Physics Derivation Graph. Eq. \ref{eq:1292901} is an initial equation. $$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0 \label{eq:1292901}$$ Change variable $$y$$ to $$y_f$$ and $$t$$ to $$t_f$$ in Eq. \ref{eq:1292901}; yields Eq. \ref{eq:8592617}. $$y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 \label{eq:8592617}$$ Boundary condition: Eq. \ref{eq:4911015} when Eq. \ref{eq7946350}. y(t_f) = y_f = 0 $$y_f = 0 \label{eq:4911015}$$ LHS of Eq. \ref{eq:8592617} is equal to LHS of Eq. \ref{eq:4911015}; yields Eq. \ref{eq:7336772}. $$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0 \label{eq:7336772}$$ Eq. \ref{eq:2601896} is an assumption. $$y_0 = 0 \label{eq:2601896}$$ Substitute LHS of Eq. \ref{eq:7336772} into Eq. \ref{eq:2601896}; yields Eq. \ref{eq:7465542}. $$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) \label{eq:7465542}$$ Divide both sides of Eq. \ref{eq:7465542} by $$t_f$$; yields Eq. \ref{eq:5115586}. $$0 = - \frac{1}{2} g t_f + v_0 \sin(\theta) \label{eq:5115586}$$ Add $$\frac{1}{2} g t_f$$ to both sides of Eq. \ref{eq:5115586}; yields Eq. \ref{eq:3904454}. $$\frac{1}{2} g t_f = v_0 \sin(\theta) \label{eq:3904454}$$ Multiply both sides of Eq. \ref{eq:3904454} by $$2/g$$; yields Eq. \ref{eq:8982886}. $$t_f = \frac{2 v_0 \sin(\theta)}{g} \label{eq:8982886}$$ Eq. \ref{eq:2022953} is an initial equation. $$x = v_0 t \cos(\theta) + x_0 \label{eq:2022953}$$ Change variable $$x$$ to $$x_f$$ and $$t$$ to $$t_f$$ in Eq. \ref{eq:2022953}; yields Eq. \ref{eq:2293278}. $$x_f = v_0 t_f \cos(\theta) + x_0 \label{eq:2293278}$$ Boundary condition: Eq. \ref{eq:5891715} when Eq. \ref{eq1654988}. $$x_f = x_0 + d \label{eq:5891715}$$ Substitute LHS of Eq. \ref{eq:5891715} into Eq. \ref{eq:2293278}; yields Eq. \ref{eq:6742208}. $$x_0 + d = v_0 t_f \cos(\theta) + x_0 \label{eq:6742208}$$ Subtract $$x_0$$ from both sides of Eq. \ref{eq:6742208}; yields Eq. \ref{eq:6756414}. $$d = v_0 t_f \cos(\theta) \label{eq:6756414}$$ Substitute LHS of Eq. \ref{eq:8982886} into Eq. \ref{eq:6756414}; yields Eq. \ref{eq:4362314}. $$d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta) \label{eq:4362314}$$ Eq. \ref{eq:6199255} is an initial equation. $$\sin(2 x) = 2 \sin(x) \cos(x) \label{eq:6199255}$$ Change variable $$\theta$$ to $$x$$ in Eq. \ref{eq:6199255}; yields Eq. \ref{eq:7596368}. $$\sin(2 \theta) = 2 \sin(\theta) \cos(\theta) \label{eq:7596368}$$ Substitute LHS of Eq. \ref{eq:7596368} into Eq. \ref{eq:4362314}; yields Eq. \ref{eq:5129639}. $$d = \frac{v_0^2}{g} \sin(2 \theta) \label{eq:5129639}$$ The maximum of Eq. \ref{eq:5129639} with respect to $$\theta$$ is Eq. \ref{eq:2728170} $$\theta = \frac{\pi}{4} \label{eq:2728170}$$ Substitute LHS of Eq. \ref{eq:2728170} into Eq. \ref{eq:5129639}; yields Eq. \ref{eq:9834994}. $$d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right) \label{eq:9834994}$$ Simplify Eq. \ref{eq:9834994}; yields Eq. \ref{eq:6972103}. $$d = \frac{v_0^2}{g} \label{eq:6972103}$$ Eq. \ref{eq:6972103} is one of the final equations. Eq. \ref{eq:2728170} is one of the final equations.