Generated by the Physics Derivation Graph. Eq. \ref{eq:8844119} is an initial equation. \begin{equation} E = KE + PE \label{eq:8844119} \end{equation} Change of variable \(E\) to \(E_2\) and \(KE\) to \(KE_2\) and \(PE\) to \(PE_2\) in Eq. \ref{eq:8844119}; yields Eq. \ref{eq:5642407}. \begin{equation} E_2 = KE_2 + PE_2 \label{eq:5642407} \end{equation} Change of variable \(E\) to \(E_1\) and \(KE\) to \(KE_1\) and \(PE\) to \(PE_1\) in Eq. \ref{eq:8844119}; yields Eq. \ref{eq:1298003}. \begin{equation} E_1 = KE_1 + PE_1 \label{eq:1298003} \end{equation} Subtract Eq. \ref{eq:1298003} from Eq. \ref{eq:5642407}; yields Eq. \ref{eq:5642407}. \begin{equation} E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1) \label{eq:2443387} \end{equation} Eq. \ref{eq:6559987} is an initial equation. \begin{equation} KE = \frac{1}{2} m v^2 \label{eq:6559987} \end{equation} Change variable \(KE\) to \(KE_2\) and \(v\) to \(v_2\) in Eq. \ref{eq:6559987}; yields Eq. \ref{eq:6632540}. \begin{equation} KE_2 = \frac{1}{2} m v_2^2 \label{eq:6632540} \end{equation} Change variable \(KE\) to \(KE_1\) and \(v\) to \(v_1\) in Eq. \ref{eq:6559987}; yields Eq. \ref{eq:4208138}. \begin{equation} KE_1 = \frac{1}{2} m v_1^2 \label{eq:4208138} \end{equation} Subtract Eq. \ref{eq:4208138} from Eq. \ref{eq:6632540}; yields Eq. \ref{eq:6632540}. \begin{equation} KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right) \label{eq:9602854} \end{equation} Divide both sides of Eq. \ref{eq:9602854} by \(t\); yields Eq. \ref{eq:3040361}. \begin{equation} \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t} \label{eq:3040361} \end{equation} Eq. \ref{eq:2776565} is an initial equation. \begin{equation} x^2 - y^2 = (x+y)(x-y) \label{eq:2776565} \end{equation} Change variable \(y\) to \(v_1\) and \(x\) to \(v_2\) in Eq. \ref{eq:2776565}; yields Eq. \ref{eq:8696678}. \begin{equation} v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1) \label{eq:8696678} \end{equation} Substitute RHS of Eq. \ref{eq:8696678} into Eq. \ref{eq:3040361}; yields Eq. \ref{eq:6246951}. \begin{equation} \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t} \label{eq:6246951} \end{equation} Eq. \ref{eq:3484339} is an initial equation. \begin{equation} v = \frac{v_1 + v_2}{2} \label{eq:3484339} \end{equation} Eq. \ref{eq:6973462} is an initial equation. \begin{equation} a = \frac{v_2 - v_1}{t} \label{eq:6973462} \end{equation} Substitute RHS of Eq. \ref{eq:3484339} into Eq. \ref{eq:6246951}; yields Eq. \ref{eq:6733685}. \begin{equation} \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t} \label{eq:6733685} \end{equation} Substitute RHS of Eq. \ref{eq:6973462} into Eq. \ref{eq:6733685}; yields Eq. \ref{eq:4876963}. \begin{equation} \frac{KE_2 - KE_1}{t} = m v a \label{eq:4876963} \end{equation} Eq. \ref{eq:8447573} is an initial equation. \begin{equation} F = m a \label{eq:8447573} \end{equation} Substitute RHS of Eq. \ref{eq:8447573} into Eq. \ref{eq:4876963}; yields Eq. \ref{eq:7034924}. \begin{equation} \frac{KE_2 - KE_1}{t} = v F \label{eq:7034924} \end{equation} Eq. \ref{eq:8497204} is an initial equation. \begin{equation} PE = -F x \label{eq:8497204} \end{equation} Change variable \(PE\) to \(PE_2\) and \(x\) to \(x_2\) in Eq. \ref{eq:8497204}; yields Eq. \ref{eq:3988671}. assumes constant force \begin{equation} PE_2 = -F x_2 \label{eq:3988671} \end{equation} Change variable \(PE\) to \(PE_1\) and \(x\) to \(x_1\) in Eq. \ref{eq:8497204}; yields Eq. \ref{eq:9081932}. \begin{equation} PE_1 = -F x_1 \label{eq:9081932} \end{equation} Subtract Eq. \ref{eq:9081932} from Eq. \ref{eq:3988671}; yields Eq. \ref{eq:3988671}. \begin{equation} PE_2 - PE_1 = -F ( x_2 - x_1 ) \label{eq:1550851} \end{equation} Divide both sides of Eq. \ref{eq:1550851} by \(t\); yields Eq. \ref{eq:7539016}. \begin{equation} \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right) \label{eq:7539016} \end{equation} Eq. \ref{eq:6154610} is an initial equation. \begin{equation} v = \frac{x_2 - x_1}{t} \label{eq:6154610} \end{equation} Substitute RHS of Eq. \ref{eq:6154610} into Eq. \ref{eq:7539016}; yields Eq. \ref{eq:9383749}. \begin{equation} \frac{PE_2 - PE_1}{t} = -F v \label{eq:9383749} \end{equation} Divide both sides of Eq. \ref{eq:2443387} by \(t\); yields Eq. \ref{eq:2692856}. \begin{equation} \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t} \label{eq:2692856} \end{equation} Substitute RHS of Eq. \ref{eq:9383749} into Eq. \ref{eq:2692856}; yields Eq. \ref{eq:9714818}. \begin{equation} \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v \label{eq:9714818} \end{equation} Substitute RHS of Eq. \ref{eq:7034924} into Eq. \ref{eq:9714818}; yields Eq. \ref{eq:5300304}. \begin{equation} \frac{E_2 - E_1}{t} = v F - F v \label{eq:5300304} \end{equation} Simplify Eq. \ref{eq:5300304}; yields Eq. \ref{eq:6495233}. \begin{equation} \frac{E_2 - E_1}{t} = 0 \label{eq:6495233} \end{equation} Multiply both sides of Eq. \ref{eq:6495233} by \(t\); yields Eq. \ref{eq:2075807}. \begin{equation} E_2 - E_1 = 0 \label{eq:2075807} \end{equation} Add \(E_1\) to both sides of Eq. \ref{eq:2075807}; yields Eq. \ref{eq:1781127}. \begin{equation} E_2 = E_1 \label{eq:1781127} \end{equation} Eq. \ref{eq:1781127} is one of the final equations.