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?
Best Answer
More than that. It is not possible to have any collision in which they are not preserved.
However, you are not completely off-base here. Let's think about just what we mean by saying that "energy is not conserved" in a inelastic collision in the first place.
We don't actually mean that energy disappears, we mean that it disappears from the bulk kinetic terms (i.e. $\frac{1}{2} m_a v^2_{a1}$ and it's friends), and ends up in some other forms that we don't consider in our kinematics (mostly sound and heat for classroom demos).
Likewise some of the angular momentum could disappear from the bulk terms like $\vec{v}_{a1} \times m_a \vec{p}_{a1}$ into a channel that you are not writing down: the internal angular momentum of the product mass(es).
To be completely correct you should be attaching a moment of inertia and a angular velocity figure to each of your masses.
Next question for a simulation. When to almost-but-not-quite-point-masses with moments of inertia $I_{a,b}$ collide and stick what should the moment of inertia $I$ of the combined mass be and why?