Generated by the Physics Derivation Graph. Eq. \ref{eq:6758737} is an initial equation. \begin{equation} \vec{a} = \frac{d\vec{v}}{dt} \label{eq:6758737} \end{equation} Substitute LHS of Eq. \ref{eq:6758737} into Eq. \ref{eq:4862823}; yields Eq. \ref{eq:4755350}. \begin{equation} a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt} \label{eq:4755350} \end{equation} Substitute LHS of Eq. \ref{eq:5359560} into Eq. \ref{eq:4755350}; yields Eq. \ref{eq:4904941}. \begin{equation} a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right) \label{eq:4904941} \end{equation} Separate two vector components in Eq. \ref{eq:4904941}; yields Eq. \ref{eq:5765841} and Eq. \ref{eq:2080932} \begin{equation} a_x = \frac{d}{dt} v_x \label{eq:5765841} \end{equation} \begin{equation} a_y = \frac{d}{dt} v_y \label{eq:2080932} \end{equation} Eq. \ref{eq:2060958} is an assumption. define the orientation of the coordinate system with respect to the gravitational acceleration such that x axis is perpendicular to gravity \begin{equation} a_x = 0 \label{eq:2060958} \end{equation} Eq. \ref{eq:1439312} is an assumption. define the orientation of the coordinate system with respect to the gravitational acceleration such that y axis is parallel to gravity \begin{equation} a_y = -g \label{eq:1439312} \end{equation} Assume \(2\) dimensions; decompose vector to be Eq. \ref{eq:4862823}. \begin{equation} \vec{a} = a_x \hat{x} + a_y \hat{y} \label{eq:4862823} \end{equation} Assume \(2\) dimensions; decompose vector to be Eq. \ref{eq:5359560}. \begin{equation} \vec{v} = v_x \hat{x} + v_y \hat{y} \label{eq:5359560} \end{equation} Substitute LHS of Eq. \ref{eq:2060958} into Eq. \ref{eq:5765841}; yields Eq. \ref{eq:8742281}. \begin{equation} 0 = \frac{d}{dt} v_x \label{eq:8742281} \end{equation} Substitute LHS of Eq. \ref{eq:1439312} into Eq. \ref{eq:2080932}; yields Eq. \ref{eq:3939933}. \begin{equation} -g = \frac{d}{dt} v_y \label{eq:3939933} \end{equation} Multiply both sides of Eq. \ref{eq:3939933} by \(dt\); yields Eq. \ref{eq:4777195}. \begin{equation} -g dt = d v_y \label{eq:4777195} \end{equation} Indefinite integral of both sides of Eq. \ref{eq:4777195}; yields Eq. \ref{eq:3366698}. \begin{equation} -g \int dt = \int d v_y \label{eq:3366698} \end{equation} Simplify Eq. \ref{eq:3366698}; yields Eq. \ref{eq:1321587}. \begin{equation} -g t = v_y - v_{0, y} \label{eq:1321587} \end{equation} Add \(v_{0, y}\) to both sides of Eq. \ref{eq:1321587}; yields Eq. \ref{eq:2682139}. \begin{equation} -g t + v_{0, y} = v_y \label{eq:2682139} \end{equation} Eq. \ref{eq:3936380} is an initial equation. \begin{equation} v_y = \frac{dy}{dt} \label{eq:3936380} \end{equation} Substitute LHS of Eq. \ref{eq:3936380} into Eq. \ref{eq:2682139}; yields Eq. \ref{eq:5010170}. \begin{equation} -g t + v_{0, y} = \frac{dy}{dt} \label{eq:5010170} \end{equation} Multiply both sides of Eq. \ref{eq:5010170} by \(dt\); yields Eq. \ref{eq:5577963}. \begin{equation} -g t dt + v_{0, y} dt = dy \label{eq:5577963} \end{equation} Indefinite integral of both sides of Eq. \ref{eq:5577963}; yields Eq. \ref{eq:8020644}. \begin{equation} -g \int t dt + v_{0, y} \int dt = \int dy \label{eq:8020644} \end{equation} Simplify Eq. \ref{eq:8020644}; yields Eq. \ref{eq:8638087}. \begin{equation} - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0 \label{eq:8638087} \end{equation} Add \(y_0\) to both sides of Eq. \ref{eq:8638087}; yields Eq. \ref{eq:7541692}. \begin{equation} - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y \label{eq:7541692} \end{equation} Multiply both sides of Eq. \ref{eq:8742281} by \(dt\); yields Eq. \ref{eq:5239397}. \begin{equation} 0 dt = d v_x \label{eq:5239397} \end{equation} Indefinite integral of both sides of Eq. \ref{eq:5239397}; yields Eq. \ref{eq:3137944}. \begin{equation} \int 0 dt = \int d v_x \label{eq:3137944} \end{equation} Simplify Eq. \ref{eq:3137944}; yields Eq. \ref{eq:9737190}. \begin{equation} 0 = v_x - v_{0, x} \label{eq:9737190} \end{equation} Add \(v_{0, x}\) to both sides of Eq. \ref{eq:9737190}; yields Eq. \ref{eq:8435615}. \begin{equation} v_{0, x} = v_x \label{eq:8435615} \end{equation} Eq. \ref{eq:4895553} is an initial equation. \begin{equation} v_x = \frac{dx}{dt} \label{eq:4895553} \end{equation} Substitute LHS of Eq. \ref{eq:8435615} into Eq. \ref{eq:4895553}; yields Eq. \ref{eq:5123314}. \begin{equation} v_{0, x} = \frac{dx}{dt} \label{eq:5123314} \end{equation} Multiply both sides of Eq. \ref{eq:5123314} by \(dt\); yields Eq. \ref{eq:8062944}. \begin{equation} v_{0, x} dt = dx \label{eq:8062944} \end{equation} Indefinite integral of both sides of Eq. \ref{eq:8062944}; yields Eq. \ref{eq:2732393}. \begin{equation} v_{0, x} \int dt = \int dx \label{eq:2732393} \end{equation} Simplify Eq. \ref{eq:2732393}; yields Eq. \ref{eq:2740672}. \begin{equation} v_{0, x} t = x - x_0 \label{eq:2740672} \end{equation} Add \(x_0\) to both sides of Eq. \ref{eq:2740672}; yields Eq. \ref{eq:6277762}. \begin{equation} v_{0, x} t + x_0 = x \label{eq:6277762} \end{equation} Swap LHS of Eq. \ref{eq:6277762} with RHS; yields Eq. \ref{eq:3011802}. \begin{equation} x = v_{0, x} t + x_0 \label{eq:3011802} \end{equation} Assume \(2\) dimensions; decompose vector to be Eq. \ref{eq:1381925}. \begin{equation} \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y} \label{eq:1381925} \end{equation} Separate vector in Eq. \ref{eq:1381925} into components related by angle \(\theta\); yields Eq. \ref{eq:5523081} and Eq. \ref{eq:2378061}. \begin{equation} \cos(\theta) = \frac{v_{0, x}}{v_0} \label{eq:5523081} \end{equation} \begin{equation} \sin(\theta) = \frac{v_{0, y}}{v_0} \label{eq:2378061} \end{equation} Multiply both sides of Eq. \ref{eq:5523081} by \(v_0\); yields Eq. \ref{eq:6010171}. \begin{equation} v_0 \cos(\theta) = v_{0, x} \label{eq:6010171} \end{equation} Substitute LHS of Eq. \ref{eq:6010171} into Eq. \ref{eq:3011802}; yields Eq. \ref{eq:6795282}. \begin{equation} x = v_0 t \cos(\theta) + x_0 \label{eq:6795282} \end{equation} Eq. \ref{eq:6795282} is one of the final equations. Multiply both sides of Eq. \ref{eq:2378061} by \(v_0\); yields Eq. \ref{eq:3041148}. \begin{equation} v_0 \sin(\theta) = v_{0, y} \label{eq:3041148} \end{equation} Swap LHS of Eq. \ref{eq:7541692} with RHS; yields Eq. \ref{eq:1910429}. \begin{equation} y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0 \label{eq:1910429} \end{equation} Substitute LHS of Eq. \ref{eq:3041148} into Eq. \ref{eq:1910429}; yields Eq. \ref{eq:9780510}. \begin{equation} y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0 \label{eq:9780510} \end{equation} Eq. \ref{eq:9780510} is one of the final equations.