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@ -2,12 +2,8 @@ $$ m_{1}\vec{v}_{1} + m_{2}\vec{v}_{2} = m_{1}\vec{v'}_{1} + m_{2}\vec{v'}_{2} $
$$ \frac{m_{1}{v_{1}}^2}{2} + \frac{m_{2}{v_{2}}^2}{2} = \frac{m_{1}{v'_{1}}^2}{2} + \frac{m_{2}{v'_{2}}^2}{2} $$ $$ \frac{m_{1}{v_{1}}^2}{2} + \frac{m_{2}{v_{2}}^2}{2} = \frac{m_{1}{v'_{1}}^2}{2} + \frac{m_{2}{v'_{2}}^2}{2} $$
$$ \left\{ $$\begin{cases}m_{1}({v_{1}} - {v'_{1}}) = m_{2}({v'_{2}} - {v_{2}})\\m_{1}({v_{1}}^2 - {v'_{1}}^2) = m_{2}({v'_{2}}^2 - {v_{2}}^2)
\begin{array}{lr} \end{cases}$$
m_{1}({v_{1}} - {v'_{1}}) = m_{2}({v'_{2}} - {v_{2}})\\
m_{1}({v_{1}}^2 - {v'_{1}}^2) = m_{2}({v'_{2}}^2 - {v_{2}}^2)
\end{array}
\right. $$
$$ \frac{v_{1} - v'_{1}}{{v_{1}^2} - {v'_{1}}^2} = \frac{v'_{2} - v_{2}}{{{v'_{2}}^2} - {v_{2}}^2} $$ $$ \frac{v_{1} - v'_{1}}{{v_{1}^2} - {v'_{1}}^2} = \frac{v'_{2} - v_{2}}{{{v'_{2}}^2} - {v_{2}}^2} $$
@ -17,23 +13,17 @@ $$ \frac{1}{v_1 + v'_1} = \frac{1}{v'_2 + v_2} $$
$$ v_1 + v'_1 = v'_2 + v_2 $$ $$ v_1 + v'_1 = v'_2 + v_2 $$
$$ \left\{ $$\begin{cases}
\begin{array}{lr}
v'_1 = v_2 + v'_2 - v_1\\ v'_1 = v_2 + v'_2 - v_1\\
v'_2 = v_1 + v'_1 - v_2 v'_2 = v_1 + v'_1 - v_2
\end{array} \end{cases}$$
\right. $$
$$ \left\{ $$ \begin{cases}
\begin{array}{lr}
m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2)\\ m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2)\\
m_1v_1 + m_2v_2 = m_1(v_2 + v'_2 - v_1) + m_2v'_2 m_1v_1 + m_2v_2 = m_1(v_2 + v'_2 - v_1) + m_2v'_2
\end{array} \end{cases} $$
\right. $$
$$ \left\{ $$ \begin{cases}
\begin{array}{lr}
v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2}\\ v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2}\\
v'_2 = \frac{2m_1v_1 + v_2(m_2-m_1)}{m_1+m_2} v'_2 = \frac{2m_1v_1 + v_2(m_2-m_1)}{m_1+m_2}
\end{array} \end{cases} $$
\right. $$