Editing Inelastic collision (section)

==Formula==

The formula for the velocities after a one-dimensional collision is:
<math display=block>
\begin{align}
v_a &= \frac{C_R m_b (u_b - u_a) + m_a u_a + m_b u_b} {m_a+m_b} \\
v_b &= \frac{C_R m_a (u_a - u_b) + m_a u_a + m_b u_b} {m_a+m_b}
\end{align}
</math>

where

*''v''<sub>a</sub> is the final velocity of the first object after impact
*''v''<sub>b</sub> is the final velocity of the second object after impact
*''u''<sub>a</sub> is the initial velocity of the first object before impact
*''u''<sub>b</sub> is the initial velocity of the second object before impact
*''m''<sub>a</sub> is the mass of the first object
*''m''<sub>b</sub> is the mass of the second object
*''C''<sub>R</sub> is the [[coefficient of restitution]]; if it is 1 we have an [[elastic collision]]; if it is 0 we have a perfectly inelastic collision, see below.

In a [[center of momentum frame]] the formulas reduce to:

<math display=block>
\begin{align}
v_a &= -C_R u_a \\
v_b &= -C_R u_b
\end{align}
</math>

For two- and three-dimensional collisions the velocities in these formulas are the components perpendicular to the tangent line/plane at the point of contact.

If assuming the objects are not rotating before or after the collision, the [[Normal (geometry)|normal]] [[Impulse (physics)|impulse]] is:

<math display=block>J_{n} = \frac{m_{a} m_{b}}{m_{a} + m_{b}} (1 + C_R) (\vec{u_{b}} - \vec{u_{a}}) \cdot \vec{n}</math>

where <math>\vec{n}</math> is the normal vector.

Assuming no friction, this gives the velocity updates:

<math display=block>
\begin{align}
\Delta \vec{v_{a}} &= \frac{J_{n}}{m_{a}} \vec{n} \\
\Delta \vec{v_{b}} &= -\frac{J_{n}}{m_{b}} \vec{n}
\end{align}
</math>