Elastic collisions between multiple bodies

[Lostinthought]

When considering the case of two or more point masses colliding in a 2 dimensional plane, is there any way to determine the final state completely from the initial conditions? if not is it not a blow to the deterministic ways of classical physics...

 

[hms.tech]

Well, i am not sure what u mean by the final state of the point masses .
If you mean the final velocities, then u can use these relations :
Newton's law of restitution : U(1) - U(2) = -V(1) + V(2)
and
the consevation of Kinetic Energy : K.E(before) = K.E(after)

However, i am not quite sure about two or more particles, i mean...how does the law of restituion apply here.

 

[Lostinthought]

The problem is like this, i have the initial velocities and masses, i want to find the final velocities. I have assumed elastic collision , so i have 2 momentum equations and an energy equation. But i have 4 variables, I have not used the restitution formula, because i have assumed elastic collision and have balanced the kinetic energies, am i right? or is something missing?

 

[hms.tech]

As i said, i am not quite sure about the law of restitution, i think it may be applicable in this way :
If u are saying that u have 4 variables this implies that u are dealing with Four Particles . Am i right ?

If yes, then my guess is that :

1st eq : conservation of Kinetic energies
2nd eq : Newton's law of restitution for particle 1 and 2
3rd eq : Newton's law of restitution for particle 1 and 3
4th eq : Newton's law of restitution for particle 1 and 4

4 equations, 4 variables.

 

[Lostinthought]

I have only two,the 4 variables are the x and y components of each of the particles, is using the KE balance same as using the restitution formula with e=1?, From where does the law of restitution come?, it is not a fundamental conservation law, so what is the principle behind it?. And while dealing with point masses, what direction do i take as the common normal to apply the restitution formula?

 

[maimonides]

You will have to apply momentum conservation (momentum is a vector, so angles will come into this) and energy conservation to solve this. And you´d better start with two particles.
I did not know that Newton posited a "law of restitution".

 

[Lostinthought]

What is the concept behind it?, why should coefficient of restitution be 1 for elastic collisions, can it be derived in some way?

 

[maimonides]

http://en.wikipedia.org/wiki/Restitution_coefficient
CoR = 1 is the definition of an elastic collision.

 

[jtbell]

When considering the case of two or more point masses colliding in a 2 dimensional plane, is there any way to determine the final state completely from the initial conditions?

No, there isn't.

if not is it not a blow to the deterministic ways of classical physics...

No, because real objects are not point masses.

For example, consider two circular disks sliding on a frictionless table. Let them collide elastically, and without any friction between them (to keep things simple). In addition to the x- and y-velocity and the mass of each disk, we now also have as an initial condition the "impact parameter," the distance of the center of one disk from the line of motion of the other one, at impact. (It measures how "glancing" or "head-on" the collision is.)

From the geometry of the collision, we can get an equation that includes the impact parameter and which constrains the final velocity components. We now have four equations for four unknowns (the final x- and y-components of velocity for each disk) and can solve for the final state.

1. The constraint equation.
2. Conservation of total x-component of momentum.
3. Conservation of total y-component of momentum.
4. Conservation of total (kinetic) energy.

 

[D H]

No, there isn't.

And the reason there isn't is because the differential equations are not Lipschitz continuous for this non-real problem where point masses don't interact unless they are touching, and how they interact is underspecified. There are other examples where the differential equations are not Lipschitz continuous and hence there isn't a unique solution. Norton's dome, collections of point masses interacting gravitationally that under just the right conditions don't collide but yet send particles off to infinity in finite time, supertasks involving an infinite number of point masses arranged just so, etc.

The places where Newtonian mechanics fails to be deterministic are either non-realistic (point masses that don't interact unless they are touching) or are a space of measure zero.