Elastic & Inelastic Collisions: Multiple Coefficents of Restitution ?

I've recently been looking for information on how to describe elastic and inelastic collisions in a way that spans the full set of possibility of different levels of elasticity between objects.

In both of my physics textbooks I was only able to find quantified information on the two extremes, perfectly inelastic and perfectly elastic. Eventually I found an equation that seems to span the full range between them.

I found it on Wikipedia actually, should have looked there earlier. Here's the link:

http://en.wikipedia.org/wiki/Inelastic_collision" [Broken]

I'm referring to the first two equations, Va and Vb.

My concern with the given equations is that the pair contains only one coefficient of restitution (i.e. only one measure of elasticity). In the context in which I want to do collisions, I want to specify a particular elasticity for every object in the system.

Thus, if two objects in my system collide then I have two measures of elasticity but the equation I've found for working out the collisions only has one. So, my question is, how do I correctly implement both of the elasticitys of the objects?

I've been trying to think about it by example to see if I can derive it myself.

For example, suppose a steel ball (nearly perfectly elastic) collides with mud (nearly perfectly inelastic). In that case the steel ball should get stuck to the mud and most of the energy will get transferred to deformation of the mud. In other words, it would behave like an inelastic collision despite the steel ball's elasticity. This implies that inelasticity is more of a dominate factor than elasticity is in collisions, in some sense.

My problem is I don't know how to extend this to the more general cases to correctly model how different values of elasticity interact with each other during collision.

What's the general solution? Will each object be effected by different elasticities or will it be merged into one value Cr applied to both somehow?
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I believe my previous example of the steel ball colliding with mud was perhaps overly simplistic. I have been contemplating it further and it seems that the coefficient of restitution (Cr) is effected by more than just the elasticity of each object.

All of the following characteristics of each object seem relevant to calculating the nature of the coefficient of restitution between the objects:

1. Elasticity - a measure of an object's tendency to return to it's original shape after being deformed

2. Deformability - a measure of how easily an object is deformed, essentially rigidness (what's the official term for this?)

3. Mass - because mass effects an object's ability to deform another, e.g. light objects may reflect off of objects that other heavier objects would deform

4. Surface texture / friction - because for example a sticky or rough surface may grab onto another during instantaneous collision. The different values for each object also interact, e.g. a polished steel ball hitting a ball covered in velcro will not stick but a tennis ball might

For simplicity let's assume we're talking about uniform material objects that are perfectly spherical in shape and have uniform texture as well, and collide in a vacuum, and only two objects at a time.

For reference purposes it would be nice to have an accurate table of coefficients of restitution for many different pairings of object material types. Anyone know of a good one?

I've read that even things like temperature can effect collision behavior. Also, I know that collisions speed can too, such as if you shoot a bullet fast enough at a pool of water the water can't move aside fast enough and the bullet can't penetrate it (acts solid).

I'm looking for a reasonably simple model, but a simple representation of temperature and speed phenomena would be useful as well.

Anyway, let's mention some more examples illustrating the effect of the new parameters I listed before:

A. If you drop a steel ball against a wooden floor the ball will not bounce much. A steel ball has very high elasticity but is also very rigid (unwilling to deform easily). The wood floor deforms more easily than the steel. Thus a large portion of the energy goes into deforming the wood and making sound etc (i.e. inelastic behavior). This seems to show that deformability must be taken into account to calculate some Cr.

B. Suppose we bounce a basketball against a wooden floor. Basketballs are highly deformable and also highly elastic. The ball hits the floor, which deforms more easily than steel, but not nearly as easily as a basketball, and thus the elasticity becomes a more predominate factor. The collision acts mostly as an elastic collision, still with a decent chunk of lost energy though, as anyone who's played basketball knows.

C. Suppose you launch a cloud of compressed air at mud. Mud is a very inelastic material. The cloud of air is very light (low mass), so when it hits the mud it will be reflected despite the mud's inelasticity, because the air particles simply don't have the momentum to deform the mud much. In a sense air is a very elastic material since if you compress and then release the compression force it will return to it's original distribution of air pressure completely (i.e. no permanent deformation).

D. Suppose you shoot a ball of water at mud. Water is very incompressable but at the same time is deformable. When it hits the mud it will deform the mud. Perhaps it is the fact that water is much more massive than air that it will deform the mud in comparison to air which didn't. Both air and water are fluid and will readily deform. Water may be incompressible but in the context of colliding with mud it has plenty of room to move. Is water "elastic"? I'm not sure. Anyone know? Perhaps it isn't, cause if you drop a water balloon on the ground it will deform it's shape but not bounce back up. Yeah, that sounds about right for simple purposes.

I find this question rather fascinating, especially since I can't seem to find any real answers to it in any sources yet. It's an interesting subject. You can't get much more physical than collisions.

The traditional way of finding the coefficient of restitution by way of pairing materials and deducing it by experiment may be fine from an experimental context, but the computation expense grows combinatorially with the number of material types and I plan to have a large number of them. The bottom line is I need a way of representing all material behaviors. This is for game development ultimately, so it doesn't strictly matter that it's not perfectly correct with respect to the real world (although it would be nice), but it must model all possible material behaviors in a believable way. Modularity is also important. Games intentionally alter the laws of physics when desirable for gameplay. Simplicity is also important. The computational complexity should be small. It is also desirable to be able to calculate the amount of momentum and energy lost to deformation and to other forms of energy, which I believe I could do easily if I understood what parts of the momentum interactions are strictly elastic and which parts are strictly inelastic, because the strictly inelastic would represent the deformation and other energy forms.

Yeah, I know it's a long post. I figured it might help to get people thinking thru more examples.

Feel free to share any thoughts on the matter, anything could help. Thanks again for your time.
Hmm... I listed mass as a parameter effecting the coefficient of restitution (Cr), but perhaps density is more relevant. Not sure.

Density is the amount of mass per unit volume, and for example an object with greater density therefore pushes with greater force per unit of it's contact area, right? Thus shouldn't an object with greater density have more ability to deform an object of lesser density?

Furthermore this makes one wonder what "deformability" is in the quantified sense. What units is deformability measured in? Is it a ratio from 0 to 1 like the Cr? Perhaps such a ratio measures the maximum percentage it can be compressed before becoming incompressible due to being already as tightly packed as it can be? Or is it dependent on other factors?

Which of my parameters are actually independent of each other to begin with?

Texture is often internal, not just on the surface. How does internal texture effect deformability? Perhaps the two are even equivalent in some sense, since internal texture is the arrangement of the materials inside the object and hence implies the manner in which substance will compress in some sense.

Too many questions, not enough constraints. I'm surprised at how difficult it's been to find this information. It seems like such an essential problem that would be covered in great detail in all standard sources, but isn't.


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The idea of "coefficient of restitution", defined in terms of the relative velocities before and after impact, is really a simple empirical definition for a particular type of collision, and (as you said) it depends on many physical parameters. The simplest way to find the coefficient for a particular situation, is measure it. Published tables of coefficients are just general guides, unless you know exactly what conditions they apply to.

It is useful in a "teaching" situation because it is more realistic than either perfectly elastic or perfectly inelastic collisions, but still simple enough so you can solve problems by hand. And of course it is also useful in some "real life" situations.

A fully "physics based" model of a collision would involve the dynamics of the bodies, with 3-dimensional stress and strain calculations (i.e. the propagation of "stress waves" through the objects), the internal energy dissipation (e.g. conversion of strain energy into heat energy), nonlnear material behaviour in solids (e.g. plasticity and/or crack propagation), and fluid dynamics for liquids (including viscosity)

All of the above topics are covered in "standard sources" at a sufficiently advanced level, but in that way of looking at the problem, you can almost think of a "collision" as just a particular instance of "nonlnear transient dynamics of fluids and solids".

Physics-based computer simulations of collisions including the above effects are carried out - for example car crash simulation, which may include not only the metal of the car but also the "mechanical properties" of the passengers, the deployment of airbags, etc.

Creating realstic computer models may take literally months, and they may take days or weeks of computing time to run. The models also need to be "tweaked" to get them to agree with the reaults of experimental crash tests on real cars. Most of that activity is commercially confidential, so it is not widely published outside of the companies who did the work.

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