diff --git a/content/blog/1d-collisions/index.md b/content/blog/1d-collisions/index.md index dcb4b57..106d539 100644 --- a/content/blog/1d-collisions/index.md +++ b/content/blog/1d-collisions/index.md @@ -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} $$ -$$ \left\{ -\begin{array}{lr} -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. $$ +$$\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) +\end{cases}$$ $$ \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 $$ -$$ \left\{ -\begin{array}{lr} +$$\begin{cases} v'_1 = v_2 + v'_2 - v_1\\ v'_2 = v_1 + v'_1 - v_2 -\end{array} -\right. $$ +\end{cases}$$ -$$ \left\{ -\begin{array}{lr} +$$ \begin{cases} m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2)\\ m_1v_1 + m_2v_2 = m_1(v_2 + v'_2 - v_1) + m_2v'_2 -\end{array} -\right. $$ +\end{cases} $$ -$$ \left\{ -\begin{array}{lr} +$$ \begin{cases} v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2}\\ v'_2 = \frac{2m_1v_1 + v_2(m_2-m_1)}{m_1+m_2} -\end{array} -\right. $$ +\end{cases} $$