Elastic collision - Part 2 - Velocity decomposition

Velocity decomposition

Let's have a close look at the collision:

When the collision occurs, the velocity vector is splitted; a part is transmitted to the other body while the rest is keeped. In the above illustration, the velocity ( $\vec{u_1}$) of body 1 is splited:

a part is transmitted to body 2 ($\vec{u_{12}}$)
a part is keeped by body 1 ($\vec{u_{11}}$)

The part transmitted is the projection along the axis passing through the center of bodies. Based on the above illustration we can write:

$$ \|\vec{u_{12}}\|=\|\vec{u_1}\|.cos(\gamma) $$

As $\gamma=\beta-\alpha$, the previous relationship becomes:

$$ \|\vec{u_{12}}\|=\|\vec{u_1}\|.cos(\beta-\alpha) $$

with $\alpha$ and $\beta$ given by:

$$ \begin{array}{r c l} \alpha &=& atan2(y_2-y_1,x_2-x_1) \\ \beta &=& atan2(y_{u_1},x_{u_1}) \end{array} $$

In conclusion:

$$ \begin{array}{r c l} \gamma_1 &=& atan2(y_{u_1},x_{u_1}) - atan2(y_2-y_1,x_2-x_1) \\ \|\vec{u_{12}}| &=& \|\vec{u_1}\|.cos(\gamma_1) \\ \|\vec{u_{11}}| &=& \|\vec{u_1}\|.sin(\gamma_1) \\ \gamma_2 &=& atan2(y_{u_2},x_{u_2}) - atan2(y_1-y_2,x_1-x_2) \\ \|\vec{u_{21}}| &=& \|\vec{u_2}\|.cos(\gamma_2) \\ \|\vec{u_{22}}| &=& \|\vec{u_2}\|.sin(\gamma_2) \end{array} $$

and

$$ \begin{array}{r c l} \alpha_1 &=& atan2(y_2-y_1,x_2-x_1) \\ \vec{u_{12}} &=& \|\vec{u_{12}}\| \left( \begin{array}{r c l} cos(\alpha_1) \\ sin(\alpha_1) \end{array} \right) \\ \vec{u_{11}} &=& \|\vec{u_{11}}\| \left( \begin{array}{r c l} -sin(\alpha_1) \\ cos(\alpha_1) \end{array} \right) \\ \alpha_2 &=& atan2(y_1-y_2,x_1-x_2) \\ \vec{u_{21}} &=& \|\vec{u_{21}}\| \left( \begin{array}{r c l} cos(\alpha_2) \\ sin(\alpha_2) \end{array} \right) \\ \vec{u_{22}} &=& \|\vec{u_{22}}\| \left( \begin{array}{r c l} -sin(\alpha_2) \\ cos(\alpha_2) \end{array} \right)
\end{array} $$

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Elastic collision - Part 2 - Velocity decomposition

Velocity decomposition

See also