How to calculate the final velocities in a perfectly elastic collision, in a 3D $n$-body simulation?

Question

As part of my project (which is testing parallel programming frameworks - I am a computer scientist), I am developing a basic, 3D $n$-body simulation. I assume that each particle is a sphere (uniform radius all around), has uniform density. Finally, there are no external forces applied to the particles, such that all collisions are perfectly elastic.

Each particle has their own, potential unique, set of the following properties:

• Mass
• Radius
• Position (x, y, z)
• Velocity (x, y, z)

So far, I have developed a solution to update the velocity & position of the particles by using Newton's Universal Law of Gravitation to determine the new velocity of each particle based on the particles around it. But this is all besides the point, I just wanted to show what I've done so far

The part I have remaining is determining the velocity of particles should they collide (I have determined whether they have collided or not). The problem is that I am have been unable to understand how to use the relevant laws of Physics (which I believe are the conservation of momentum, the conservation of kinetic energy) and solved them to obtain the final velocities of two colliding particles, given the initial velocities, masses, and distance (from either the boundary or centre of masses)

I guess what I'd find helpful is if someone could:

1. Tell me what laws are relevant in this context
2. How to solve them to get me the final velocities of both particles

This is not homework but part of a larger project. As a non-physicist or mathematician of any sort, this is out of my depth and I'd appreciate elaborate guidance. I have searched the internet for answers, but most are either for a 1D problem which I struggle to extrapolate for 3D, or are just formulas without enough context for me to understand

Answer 1

If you assume that the surfaces of the balls are frictionless, then you can assume that the force of the collisions will always act normal to the surface of the spheres at the point of contact -- which is, in turn, along the line upon which lies the sphere's centers. So you can use Newton's laws of motion here, with an impulse of known direction and unknown magnitude.

Beyond that, for any collision you have to start with the conservation of energy and the conservation of momentum. Because I'm only doing this for the frictionless case, the momentum that's conserved is just linear momentum, and you can ignore any rotation of the balls.

For each ball, then, you have a position vector $\mathbf x = \begin{bmatrix}x, y, z\end{bmatrix}$* and a velocity vector $\mathbf v = \begin{bmatrix}\dot x, \dot y, \dot z\end{bmatrix}$.

Conservation of momentum says that in the absence of any other collisions, the total momentum of any two ball system remains constant through a collision: $m_1 \mathbf v_1 + m_2 \mathbf v_2 = \mathbf p_{12}$, where $\mathbf p_{12}$ is the momentum of the two balls together, and $p_1$ and $p_2$ are their masses.

Conservation of energy says that in the absence of any other collisions, the total energy of any two ball system remains constant through a collision: $m_1 \| \mathbf v_1 \|^2 + m_2 \| \mathbf v_2 \|^2 = u$, where $\| \mathbf v \|^2 = (\dot x)^2 + (\dot y)^2 + (\dot z)^2$.

The fact that we're assuming a frictionless ball surface means that the direction of the impulse at the moment of is known, but its magnitude is unknown: the impulse on ball 1 is $\Delta \mathbf p = k\left( \mathbf x_2 - \mathbf x_1 \right)$. $\mathbf x_2 - \mathbf x_1$ gives the direction, letting $k$ be an unknown positive number captures the fact that we only know the direction of the impulse and have to let the magnitude be determined by the conservation laws.

If you take $t^-$ to be the time immediately before the collision and $t^+$ to be the time immediately after, then $\mathbf p_1(t^+) = \mathbf p_1(t^-) + \Delta \mathbf p$ and $\mathbf p_2(t^+) = \mathbf p_2(t^-) - \Delta \mathbf p$, where $\mathbf p = m \mathbf v$. Note the $+ \Delta \mathbf p$ and $- \Delta \mathbf p$, respectively.

You should at this point, have exactly enough equations in exactly enough unknowns to calculate you ending velocities from your beginning ones.

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* Apologies for the dual use of $\mathbf x$ and $x$. I hope the typeface keeps them clear.