Is it possible for two spheres (a & b) to have an inelastic collision with BOTH the total linear and angular momentum preserved? I'm doing some physics simulation of some spheres attracting each other like gravity and and a initial net angular momentum making them spin around a center. I want to have inelastic collisions while keeping the same total linear and angular momentum.
The simulation is in 3 dimensions. Therefore, all velocities are 3d vectors. I can calculate the velocity the particles after an completely inelastic collision by solving the following equations for $v$:
$$ m_a \vec v_{a_0} + m_b \vec v_{b_0} = \vec v (m_a + m_b) $$ This gives velocity (v) zero degrees of freedom, yet I have not taken into account the formula for preservation of angular momentum (around origin):
$$ m_a (\vec p_a \times \vec v_{a_0})+m_b (\vec p_b \times \vec v_{b_0}) = m_a (\vec p_a \times \vec v_{a_1})+m_b (\vec p_b \times \vec v_{b_1}) $$ And for an inelastic collision $\vec v_{a_1} = \vec v_{b_1} = \vec v$ $$ m_a (\vec p_a \times \vec v_{a_0})+m_b (\vec p_b \times \vec v_{b_0}) = \vec v \times (m_a \vec p_a+m_b \vec p_b) $$ Where $\vec p_a$ and $\vec p_b$ is the position vector of the spheres. The problem here is that the spheres also have a radius, so in the instant of a collision $p_a$ and $p_b$ is not equal. The only way I could see both angular and linear momentum being preserved is if the radius changes. Is this at all possible?