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# Elastic collision - Part 3 - Velocity calculation

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 $$\vec{v_2}$$:

$$\vec{v_2}=\vec{u_{22}}+\vec{v_{21}} = \|\vec{u_{22}}\| \left( \begin{array}{r c l} -sin(\alpha_2) \\ cos(\alpha_2) \end{array} \right) - \|\vec{v_{21}}\| \left( \begin{array}{r c l} cos(\alpha_2) \\ sin(\alpha_2) \end{array} \right)$$