Elastic collision - Part 3 - Velocity calculation

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We now focus on a sub problem: calculation of the norm of the partial velocities after the collision for each body. This problem can be seen as a head-on collision where:

$\vec{u_{12}}$ and $\vec{u_{21}}$ are the velocities before collision;
$\vec{v_{12}}$ and $\vec{v_{21}}$ are the velocities after collision.

Conservation of kinetic energy

$$ \frac{1}{2}m_1\|\vec{u_{12}}\|^2 + \frac{1}{2}m_2\|\vec{u_{21}}\|^2 = \frac{1}{2}m_1\|\vec{v_{12}}\|^2 + \frac{1}{2}m_2\|\vec{v_{21}}\|^2 $$

can be rewritten as:

$$ m_1(\|\vec{u_{12}}\|^2 - \|\vec{v_{12}}\|^2) = m_2(\|\vec{v_{21}}\|^2 - \|\vec{u_{21}}\|^2) $$

Conservation of momentum:

$$ m_1\vec{u_{12}} + m_2\vec{u_{21}} = m_1\vec{v_{12}} +m_2\vec{v_{21}} $$

can be rewritten as:

$$ m_1\|\vec{u_{12}}\|+m_2\|\vec{u_{21}}\|=m_1\|\vec{v_{12}}\|+m_2\|\vec{v_{21}}\| $$

and :

$$ m_1(\|\vec{u_{12}}\| - \|\vec{v_{12}}\|) = m_2(\|\vec{v_{21}}\|-\|\vec{u_{21}}\|) $$

Solving system of equations

Whole system is given by:

$$ \begin{cases} m_1(\|\vec{u_{12}}\|^2 &-& \|\vec{v_{12}}\|^2) & = & m_2(\|\vec{v_{21}}\|^2 &-& \|\vec{u_{21}}\|^2) \\ m_1(\|\vec{u_{12}}\| &-& \|\vec{v_{12}}\|) & = & m_2(\|\vec{v_{21}}\| &-& \|\vec{u_{21}}\|) \end{cases} $$

The previous system can be rewritten as:

$$ \begin{cases} m_1( \|\vec{u_{12}}\| + \|\vec{v_{12}}\|)( \|\vec{u_{12}}\| - \|\vec{v_{12}}\|) & = & m_2(\|\vec{v_{21}}\| - \|\vec{u_{21}}\|)(\|\vec{v_{21}}\| + \|\vec{u_{21}}\|) \\ m_1(\|\vec{u_{12}}\| - \|\vec{v_1}\|) & = & m_2(\|\vec{v_{21}}\| - \|\vec{u_{21}}\|) \end{cases} $$

Dividing the first equation by the second gives us:

$$ \|\vec{u_{12}}\| + \|\vec{v_{12}}\| = \|\vec{u_{21}}\| + \|\vec{v_{21}}\| $$

A new system can be formulated:

$$ \begin{cases} \|\vec{u_{12}}\| + \|\vec{v_{12}}\| &=& \|\vec{u_{21}}\| + \|\vec{v_{21}}\| \\ m_1(\|\vec{u_{12}}\| - \|\vec{v_{12}}\|) &=& m_2(\|\vec{v_{21}}\| - \|\vec{u_{21}}\|) \end{cases} $$

Let multiply the first equation by $m_1$:

$$ \begin{cases} m_1\|\vec{u_{12}}\| &+& m_1\|\vec{v_{12}}\| &=& m_1\|\vec{u_{21}}\| &+& m_1\|\vec{v_{21}}\| \\ m_1\|\vec{u_{12}}\| &-& m_1\|\vec{v_{12}}\| &=& m_2\|\vec{u_{21}}\| &-& m_2\|\vec{v_{21}}\| \end{cases} $$

Adding the first equation to the second gives us:

$$ 2m_1\|\vec{u_{12}}\| = (m_1-m_2)\|\vec{u_{21}}\| + (m_1+m_2)\|\vec{v_{21}}\| $$

Velocities $\|\vec{v_{12}}\|$ and $\|\vec{v_{21}}\|$ after the collision can now be deduced:

$$ \begin{cases} \|\vec{v_{12}}\| = \frac{1}{m_1+m_2} [ 2m_2\|\vec{u_{21}}\| + (m_1-m_2)\|\vec{u_{12}}\| ] \\ \|\vec{v_{21}}\| = \frac{1}{m_1+m_2} [ 2m_1\|\vec{u_{12}}\| + (m_2-m_1)\|\vec{u_{21}}\| ] \end{cases} $$

Considering opposite directions of $\vec{u_{21}}$ in regard of $\vec{u_{12}}$. The previous equations become:

$$ \begin{cases} \|\vec{v_{12}}\| = \frac{1}{m_1+m_2} [ (m_1-m_2)\|\vec{u_{12}}\| - 2m_2\|\vec{u_{21}}\| ] \\ \|\vec{v_{21}}\| = \frac{1}{m_1+m_2} [ (m_1-m_2)\|\vec{u_{21}}\| + 2m_1\|\vec{u_{12}}\| ]
\end{cases} $$

Synthesis

Velocity $\vec{v_1}$ can now be computed:

$$ \vec{v_1}=\vec{u_{11}}+\vec{v_{12}} = \|\vec{u_{11}}\| \left( \begin{array}{r c l} -sin(\alpha_1) \\ cos(\alpha_1) \end{array} \right) + \|\vec{v_{12}}\| \left( \begin{array}{r c l} cos(\alpha_1) \\ sin(\alpha_1) \end{array} \right) $$

Taking into account opposite direction of $\vec{v_{21}}$ allows us to compute $\vec{v_2}$: