Generated by the Physics Derivation Graph. Eq. \ref{eq:2237799} is an initial equation. \begin{equation} dW = F dx \label{eq:2237799} \end{equation} Integrate Eq.~ref{eq:2237799}; yields Eq.~ref{eq:2565189}. \begin{equation} \int dW = F \int_0^x dx \label{eq:2565189} \end{equation} Evaluate definite integral Eq. \ref{eq:2565189}; yields Eq. \ref{eq:7362045}. \begin{equation} W = F x \label{eq:7362045} \end{equation} Substitute LHS of Eq. \ref{eq:7362045} into Eq. \ref{eq:3086821}; yields Eq. \ref{eq:6167182}. \begin{equation} W = m a x \label{eq:6167182} \end{equation} Change of variable \(d\) to \(x\) and \(v\) to \(v_2\) and \(v\) to \(v_1\) in Eq. \ref{eq:4741344}; yields Eq. \ref{eq:5997798}. \begin{equation} x = \frac{v_2^2 - v_1^2}{2 a} \label{eq:5997798} \end{equation} Substitute LHS of Eq. \ref{eq:5997798} into Eq. \ref{eq:6167182}; yields Eq. \ref{eq:6760874}. \begin{equation} W = m a \frac{v_2^2 - v_1^2}{2 a} \label{eq:6760874} \end{equation} Simplify Eq. \ref{eq:6760874}; yields Eq. \ref{eq:4236963}. \begin{equation} W = \frac{1}{2} m v_2^2 - \frac{1}{2} m v_1^2 \label{eq:4236963} \end{equation} Substitute LHS of Eq. \ref{eq:8207477} and LHS of Eq. \ref{eq:8883350} into Eq. \ref{eq:4236963}; yields Eq. \ref{eq:4943050}. \begin{equation} W = KE_2 - KE_1 \label{eq:4943050} \end{equation} Eq. \ref{eq:4943050} is one of the final equations.