3D sphere oblique impacts calculations

[Logan Blinco]

Hello,

Im creating a physics simulator and I am struggling to expand my collisions from 2D to 3D. In 2D the velocity only changes parallel to the line of center so I presume this is the same for 3D.I can get a Cartesian equation of line but I am not sure how to get the velocity component relative to this line.

So my question is how can I create a relative axis which would let me use linear momentum conservation or more generically how to Calculate the final velocity of 3D sphere collisions.

 

[DrStupid]

It depends: Rigid or not rigid? Elastic or inelastic? Friction or no friction? ...

 

[Logan Blinco]

It depends: Rigid or not rigid? Elastic or inelastic? Friction or no friction? ...

Rigid.Using coefficient of Restitution and no friction.

 

[DrStupid]

Rigid.Using coefficient of Restitution and no friction.

OK, let's say p is the momentum of one of the two sphere in their common rest frame. Than the corresponding initial total kinetic energy is

$E = \frac{{p^2 }}{2} \cdot \left( {\frac{1}{{m_1 }} + \frac{1}{{m_1 }}} \right)$

During the collision the the momentum

$\Delta p = k \cdot \Delta r$

is exchanged, where $\Delta r$ is the distance between the spheres. This results in the final kinetic energy

$E' = \frac{{\left( {p + k \cdot \Delta r} \right)^2 }}{2} \cdot \left( {\frac{1}{{m_1 }} + \frac{1}{{m_1 }}} \right) = e^2 \cdot E$

where e is the coefficient of restitution. Now you just need to solve for k and and calculate the resulting velocities.