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