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