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