Spherical Collisions: Solving Non-Equal Mass/Radius

[Falmarri]

Is there a solution to finding out how exactly 2 spheres of non-equal mass and radius rebound after they collide? I know how to do it for equal mass and radius, finding the line of the collision, and then the components of velocity perpendicular to that line don't change. Does that hold true for all spherical collisions or just uniform spheres?

 

[SystemTheory]

Standard collision examples:

http://hyperphysics.phy-astr.gsu.edu/Hbase/colsta.html

 

[Falmarri]

I just took a look at that. That doesn't seem to have a solution for when the spheres are offset. For example if sphere 1 is 1 meter, sphere 2 is 3 meters. Sphere one is at x=0 with velocity y=2. Sphere two is at x=1 with velocity y=-2.

 

[dE_logics]

Is there a solution to finding out how exactly 2 spheres of non-equal mass and radius rebound after they collide? I know how to do it for equal mass and radius, finding the line of the collision, and then the components of velocity perpendicular to that line don't change. Does that hold true for all spherical collisions or just uniform spheres?

IMO under ideal situations where you do not consider the rotation of the sphere, the diameter should not matter.

Rest you can figure out the final velocity of the 2 balls (but only assuming the above is true) using conservation of kinetic energy and momentum.

You need to make 2 sets of equations from fact that K.E and momentum of the arrangement will remain the same after collision (one from momentum and one from the K.E.).

 

[Andy Resnick]

Unless you ignore or completely specify the mechanical properties of the spheres (and any interaction between the two other than direct contact), you cannot completely solve the problem.

 

[SystemTheory]

To see if there is a geometric interpretation I would sketch the problem with at the point of contact. Draw a radius on each sphere (drawn as a circle in a plane) to the contact point, and put up a velocity vector on each sphere. The impulse force should act along the common radius and have x-y components. It may be possible to treat the spheres as perfectly elastic and resolve the momentum components. These are just my first thoughts on a possible approach ...

 

[Falmarri]

I posted this on a programming forum as well, as that's the end result of this problem. But here is where I'm at; am I correct so far?



I'm working on the math now. And this is what I have, am I on the right track?

Ball 1 has mass M1 and is at p1=[2,5] with v1 = [-1,0], ball 2 has mass M2 and is at p2=[4,1] with v2=[1,0]. Let's assume ball 1 is 3 meters and ball 2 is 2 meters (So they're colliding). Then that means the angle between the velocity of ball 2 vector and the line between the two balls is

x1 = cos theta1 = v1.(p1-p2)/root((v1.v1 * (p1-p2).(p1-p2))).

And the angle between the velocity of ball 1 vector and the line between the two balls is

x2 = cos theta2 = v2.(p2-p1)/root((v2.v2 * (p2-p1).(p2-p1))).

And so the portion of the velocity vector perpendicular to that angle is the same before and after the collision. Right? So

||v1i|| * x1 = ||v1f|| * x1

||v2i|| * x2 = ||v2f|| * x2

And since x = cos theta, sin theta =

root(1-x{1,2}^2) = y{1,2}

And M1 * || V1i || * y1 + M2 * || V2i || * y2 = M1 * || V1f || * y1 + M2 * || V2f || * y2

Am I right so far? Now I have 1 equation for 2 unknowns, do I use KE for the other equation? I assume I break up the KE equations as well along my theta1, theta2?

(Sorry, It's been almost 5 years since I've taken mechanical physics. And I hope you can read my equations)

 

[Bob S]

If you have solid spheres of radius R and mass m which transfer angular momentum, the moment of inertia is
I = (2/5)MR²,
the angular momentum is Iw, and the rotational kinetic energy is
E = (1/2)Iw<sup>2[/SUP]
where w is the angular velocity in radians per second.

This is also important if they are rolling without slipping.

Bob S</sup>

 

[Falmarri]

I am not taking into account rotation, or friction.

 

[SystemTheory]

Physics by Halliday and Resnick says the problem of 2 or 3 dimensional collisions cannot be solved only from knowledge of initial motion of the particles, except if they stick together after the collision (totally inelastic case). This is because you will have three equations and four unknowns (two components of velocity for each body). You either need knowledge of the force interaction, or knowledge of one of the bodies after the collision, in addition to the initial motion, to solve such problems.

I'm pretty good at shooting pool (pocket billiards) so I wonder how my body anticipates the result so well?

 

[Falmarri]

If the solution is unsolvable, what approximations are used to do this kind of simulation in software? Games, etc?

 

[sophiecentaur]

I seem to remember, at A level, having been given problems about colliding spheres ('billiard balls') with a coefficient of restitution of less than 1 and being able to solve them using momentum conservation and using separation velocities, modified by the restitution coefficient. I thought we did it assuming something like no change to tangential velocity (frictionless contact) and it came out alright with that approximation.
Or perhaps we always had one ball stationary to start with? But that's snooker, ain't it?

 

[SystemTheory]

My textbook says you can always pick the reference frame such that one sphere is stationary. The general problem appears to be addressed under the keyword phrase "collision detection and response" in the game developer literature. You could search for some simple tutorials and see what turns up.

 

[SystemTheory]

Is there a solution to finding out how exactly 2 spheres of non-equal mass and radius rebound after they collide? I know how to do it for equal mass and radius, finding the line of the collision, and then the components of velocity perpendicular to that line don't change. Does that hold true for all spherical collisions or just uniform spheres?

I now think the problem has a solution when b is known, but I'm unable to find a reference or derive the exact solution. Here are some interesting related resources.

Flash player simulating two-diminsional collision:

http://hypertextbook.com/facts/2006/restitution.shtml

For General Reference Billiard ball collision (2 page pdf):

http://billiards.colostate.edu/technical_proofs/TP_3-1.pdf