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Elastic collision velocity relations

To express the final velocities in terms of the initial velocities in an elastic collision, it is necessary to use the equations for the conservation of kinetic energy and conservation of momentum:

$$$(1) \quad m_1\vecphy{u}_1+m_2\vecphy{u}_2=m_1\vecphy{v}_1+m_2\vecphy{v}_2$$$ $$$(2)\quad\frac{1}{2}m_1\vecphy{u}_1^2+\frac{1}{2}m_2\vecphy{u}_2^2=\frac{1}{2}m_1\vecphy{v}_1^2+\frac{1}{2}m_2\vecphy{v}_2^2$$$

The momentum/kinetic energy lost by body 1 is equal to the momentum/kinetic energy gained by body 2: $$$ \begin{align*} (1) & \iff m_1(\vecphy{u}_1- \vecphy{v}_1)=m_2(\vecphy{v}_2-\vecphy{u}_2)\\ (2) &\ \iff \frac{1}{2}m_1(\vecphy{u}_1^2- \vecphy{v}_1^2)= \frac{1}{2}m_2(\vecphy{v}_2^2- \vecphy{u}_2^2)\\ \end{align*} $$$

Now we derive the intermediary result: $$$ (2) \iff \frac{1}{2}m_1(\vecphy{u}_1+ \vecphy{v}_1) (\vecphy{u}_1- \vecphy{v}_1)= \frac{1}{2}m_2(\vecphy{v}_2+ \vecphy{u}_2) (\vecphy{v}_2- \vecphy{u}_2) $$$ And using $$(1)$$: $$$ \begin{align*} (2) & \iff \vecphy{u}_1+ \vecphy{v}_1=\vecphy{v}_2+ \vecphy{u}_2\\ & \iff \vecphy{u}_1- \vecphy{u}_2=\vecphy{v}_2 - \vecphy{v}_1\\ & \Rightarrow \text{ relative speed of approach=relative speed of separation} \end{align*} $$$

$$\vecphy{v}_1$$ and $$\vecphy{v}_2$$ can be obtained using: $$$ \begin{align*} & m_1(\vecphy{u}_1- \vecphy{v}_1)= m_2(\vecphy{v}_2- \vecphy{u}_2)\\ \iff & \frac{m_2}{m_1}\vecphy{v}_2+ \vecphy{v}_1= \frac{m_2}{m_1}\vecphy{u}_2+ \vecphy{u}_1 \end{align*} $$$ Using $$\vecphy{u}_1- \vecphy{u}_2=\vecphy{v}_2- \vecphy{v}$$: $$$ \begin{align*} & (\frac{m_2}{m_1}+1) \vecphy{v}_2=2\vecphy{u}_1+(\frac{m_2}{m_1}-1) \vecphy{u}_2\\ \iff & \vecphy{v}_2=(\frac{2m_1}{m_1+m_2})\vecphy{u}_1+(\frac{m_2-m_1}{m_1+m_2})\vecphy{u}_2 \end{align*} $$$ A similar result holds for $$\vecphy{v}_1$$: $$$\vecphy{v}_1=(\frac{m_1-m_2}{m_1+m_2})\vecphy{u}_1+(\frac{2m_2}{m_1+m_2})\vecphy{u}_2$$$