The Coefficient of restitution refers to the speed ratio between two projectiles following an impact.
So, a perfectly elastic ball hitting the ground and bouncing up with the same speed it impacted with (obviously with a different direction) has a CoR of 1.0. And, a realistic rubber ball probably has a CoR of 0.4 or something.
However, this coefficient is a property of the rubber itself. If you took a rubber ball with a CoR of 0.4 under normal circumstances, loaded it into a hydrogen gas cannon, and slammed it into a target plate at 3000 m/s, how do you model the CoR?
It’s not going to bounce off the target plate at 0.4 * 3000 because the rubber cannot store that much energy. It’s going to hit, shatter, and some of the fragments will have a small velocity vector in the direction of the normal at the point of impact, right?
In short, the rubber material has a limited amount of energy it can hold as part of the “restoring” force. If I knew how to calculate or approximate this energy maximum, I could calculate the velocity the fragments of the ball will have when they bounce away from the target.
It depends on the transition from elastic to plastic deformation for the material - which eventually depends on how much energy is involved and the physical environment such as temperature and the target material and temperature.
So without knowing the material (and ambient conditions and target) there is no single answer to the question.
Sure. But where do I look this up? I’m also not looking for perfect answers, just something roughly right. Clearly, as you talk about ever faster collisions, the effective coefficient of restitution continues to drop.
The point is that I don’t even know what terms to look for to start on this. What term reflects the “maximum storable energy before it just shatters”? Where do I look up the elastic region?
The Coefficient of Restitution is kind of a wonky, underbgrad way of looking at colisions. I realized this when I was a wonky undergrad looking at colisions somewhat like the one you describe, where things break.
You can meaningfully use the CoR in cases where you have two well-defined objects with well-defined velocities before and after the collision. The Conservation of Momentum holds in all cases, but, depending upon how elastic the collision is, some of your kinetic energy can go into deformation. In a perfectly elastic collision, both momentum and energy are conserved and you have two equations in two unknowns (momemtum before and after collision being equal and energy before and after collision being equal – I assume you knowboth masses and both pre-collision velocities) that lets you figure out the velocities after collision. In a perfectly inelastic collision the masses stick and move off with the same velocity, so you ony need conservation of momentum to figure final velocity – but you find that energy isn’t consereved.
In real life, conditions fall somewhere between these two, and if you can know the velocities before and after collision you can calculate the coefficient of restitution. More important, tell someone the CoR and they can deduce the velocities after collision.
I was looking experimentally at cases where a system was absorbing energy, as if b a spring, then fracturing due to the collision. The situation didn’t really correspond to any sort of collision I could relate to this simple model, because the thing broke while the collision was going on. You couldn’t measure velocities after collision, because there was no single piece, and the collision between two isolated bodies never really concluded.
I suppose you could sum up the momernta and energie in all the parts of, say, a superball that fractured under collision and assign a “Coefficient of Restitution” to it. I couldn’t in my case – there was an extra spring involved, and there were too many pieces after collision. But the point is that the Coefficient of Restitution is a woefully inadequate way to describe interactions anf rfactures that take place during the time of the actually collision. It assumes that you have a relatively simple situation in which you have well-defined “before” and "after velocities, and you don’;t care about the internal details of the collision.
Mostly true, for an idealized homogenous symmetrical object and an idealized perfectly unyielding target. The farther up the energy scale you go, the less accurate these assumptions will be for any real world objects.
Not really.
See Resilience (materials science) - Wikipedia for the barest hint of what you’re talking about. Once you get past even yield strength all the *wonky undergrad *physics fails to work. (awesome terminology there CalMeacham; I’m gonna keep using it.)
Once the object absorbs more energy than it can “hold”, (cf “fracture strength”) it will fragment. The masses of the fragments will, in general, be exponentially distributed. I.e, lots of dust particles, many small fragments and a couple big hunks. Each moving at different velocities. Which velocity will be a complicated function of not only each fragment’s mass, but how and where it participated in the fragmentation process.
And each will be heated to varying extents, carrying off some non-trivial but practically incalculable fraction of the incident energy.
For hefty velocities you also need to consider the atmosphere. A 3000 m/s collision in a vacuum behaves very differently than one with an atmosphere in the middle. That speed is about Mach 9 at STP.
In an atmosphere, *lots *of interesting physics will be happening to your projectile before it even *gets *to the target. And lots more will happen at the point of collision. My WAG is you’ll get compressive heating up to high thousands of Kelvin, maybe even exceeding 10K K. So now you need to model some plasma physics as well. Here’s an interesting discussion about what happens at the point of a truly high energy collision Raindrop .
I have no doubt there are sophisticated computer models used to predict this stuff. Accurate modelling of this stuff matters in anti-missile and anti-satellite kinetic kill vehicle design, and perhaps in nuclear weapons design. And probably in ground combat vehicle armor design.
We used to have a member who worked in this area who probably could have shared at least some rules of thumb. Sadly a few years ago he was profoundly injured in a motorcycle accident. I don’t now recall whether he eventually died of his injuries, but even if still alive he’ll never post again.
Note this collision is modeled at 1/15th the velocity you’re talking about. And even at that low speed it *still *has interesting non-trivial physics going on inside the collision.
So, ok. You send an object hurtling at another object. In a vacuum. And the target object is an indestructible plate. Your object is a sphere. At low velocities, upon impact, the sphere compresses and then rebounds from the plate with stored energy.
The sphere is actually just a spring. The more compressed it is, the more energy is stored into it. And the direction it rebounds in is related to the plate’s normal.
At higher and higher velocities, there sphere gets more and more compressed with the impacts. 1/2 kx^2, more energy in the ball, greater rebound velocities.
So there’s a limit to how much compression the spring can take. Compression beyond a certain point, and the spring itself explodes. From the high speed video I’ve seen, the majority of the pieces explode on an axis perpendicular to the axis of impact.
I’d like to build a generalized model, here. I don’t need it to be empirically perfect or accurate to maybe a factor of 2 with the real thing. Any collision can be generalized as one collection of pieces slamming into another. At low speeds, the colliding objects rebound with energy stored during compression. At medium speeds, the colliding objects both get embedded in each other and apparently a cloud of small pieces gets sprayed out perpendicular to the axis of impact. At high speeds, the objects pass through each other leaving conical sprays of shrapnel destroying everything.