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https://en.wikipedia.org/wiki/Equations_of_motion
Eq. \ref{eq:7864125} is an initial equation. \begin{equation} a = \frac{v - v_0}{t} \label{eq:7864125} \end{equation} Multiply both sides of Eq. \ref{eq:7864125} by \(t\); yields Eq. \ref{eq:5666935}. \begin{equation} a t = v - v_0 \label{eq:5666935} \end{equation} Add \(v_0\) to both sides of Eq. \ref{eq:5666935}; yields Eq. \ref{eq:3386860}. \begin{equation} a t + v_0 = v \label{eq:3386860} \end{equation} Swap LHS of Eq. \ref{eq:3386860} with RHS; yields Eq. \ref{eq:8873965}. \begin{equation} v = v_0 + a t \label{eq:8873965} \end{equation} Eq. \ref{eq:8873965} is one of the final equations. Eq. \ref{eq:8658331} is an initial equation. \begin{equation} v_{\rm average} = \frac{d}{t} \label{eq:8658331} \end{equation} Eq. \ref{eq:5013638} is an initial equation. \begin{equation} v_{\rm average} = \frac{v + v_0}{2} \label{eq:5013638} \end{equation} LHS of Eq. \ref{eq:8658331} is equal to LHS of Eq. \ref{eq:5013638}; yields Eq. \ref{eq:4622149}. \begin{equation} \frac{d}{t} = \frac{v + v_0}{2} \label{eq:4622149} \end{equation} Multiply both sides of Eq. \ref{eq:4622149} by \(t\); yields Eq. \ref{eq:1476448}. \begin{equation} d = \left(\frac{v + v_0}{2}\right)t \label{eq:1476448} \end{equation} Substitute RHS of Eq. \ref{eq:8873965} into Eq. \ref{eq:1476448}; yields Eq. \ref{eq:3069767}. \begin{equation} d = \frac{(v_0 + a t) + v_0}{2} t \label{eq:3069767} \end{equation} Simplify Eq. \ref{eq:3069767}; yields Eq. \ref{eq:6881977}. \begin{equation} d = \frac{2 v_0 + a t}{2} t \label{eq:6881977} \end{equation} Simplify Eq. \ref{eq:6881977}; yields Eq. \ref{eq:5385244}. \begin{equation} d = v_0 t + \frac{1}{2} a t^2 \label{eq:5385244} \end{equation} Eq. \ref{eq:5385244} is one of the final equations. Raise both sides of Eq. \ref{eq:8873965} to \(2\); yields Eq. \ref{eq:4385757}. \begin{equation} v^2 = v_0^2 + 2 a t v_0 + a^2 t^2 \label{eq:4385757} \end{equation} Simplify Eq. \ref{eq:4385757}; yields Eq. \ref{eq:9796063}. factored 2a out of two terms \begin{equation} v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right) \label{eq:9796063} \end{equation} Substitute RHS of Eq. \ref{eq:9796063} into Eq. \ref{eq:5385244}; yields Eq. \ref{eq:7702534}. \begin{equation} v^2 = v_0^2 + 2 a d \label{eq:7702534} \end{equation} Eq. \ref{eq:7702534} is one of the final equations. Subtract \(v_0\) from both sides of Eq. \ref{eq:8873965}; yields Eq. \ref{eq:4127918}. \begin{equation} v - v_0 = a t \label{eq:4127918} \end{equation} Swap LHS of Eq. \ref{eq:4127918} with RHS; yields Eq. \ref{eq:5666935}. \begin{equation} a t = v - v_0 \label{eq:5666935} \end{equation} Divide both sides of Eq. \ref{eq:5666935} by \(a\); yields Eq. \ref{eq:8222540}. \begin{equation} t = \frac{v - v_0}{a} \label{eq:8222540} \end{equation} Substitute RHS of Eq. \ref{eq:8222540} into Eq. \ref{eq:1476448}; yields Eq. \ref{eq:9270356}. \begin{equation} d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right) \label{eq:9270356} \end{equation} Simplify Eq. \ref{eq:9270356}; yields Eq. \ref{eq:7103968}. difference of squares \begin{equation} d = \frac{1}{2 a} (v^2 - v_0^2) \label{eq:7103968} \end{equation} Multiply both sides of Eq. \ref{eq:7103968} by \(2 a\); yields Eq. \ref{eq:6814979}. \begin{equation} 2 a d = v^2 - v_0^2 \label{eq:6814979} \end{equation} Add \(v_0^2\) to both sides of Eq. \ref{eq:6814979}; yields Eq. \ref{eq:7086842}. \begin{equation} 2 a d + v_0^2 = v^2 \label{eq:7086842} \end{equation} Swap LHS of Eq. \ref{eq:7086842} with RHS; yields Eq. \ref{eq:7702534}. \begin{equation} v^2 = v_0^2 + 2 a d \label{eq:7702534} \end{equation} Eq. \ref{eq:1476448} is one of the final equations. Subtract \(a t\) from both sides of Eq. \ref{eq:8873965}; yields Eq. \ref{eq:8007427}. \begin{equation} v - a t = v_0 \label{eq:8007427} \end{equation} Substitute RHS of Eq. \ref{eq:8007427} into Eq. \ref{eq:5385244}; yields Eq. \ref{eq:5577530}. \begin{equation} d = (v - a t) t + \frac{1}{2} a t^2 \label{eq:5577530} \end{equation} Simplify Eq. \ref{eq:5577530}; yields Eq. \ref{eq:8442394}. \begin{equation} d = v t - a t^2 + \frac{1}{2} a t^2 \label{eq:8442394} \end{equation} Simplify Eq. \ref{eq:8442394}; yields Eq. \ref{eq:3917794}. \begin{equation} d = v t - \frac{1}{2} a t^2 \label{eq:3917794} \end{equation} Eq. \ref{eq:3917794} is one of the final equations.