Collision between hard spheres

[dorker]

The wikipedia article on collisions says "collisions between hard spheres may be nearly elastic". It doesn't elaborate nor give a source on the statement. I was wondering about those two conditions. Hardness I understand, but is the spherical shape necessary? Is it about there being a single point of contact? In which case, what if they were non-spherical bodies, but with spherical points of collision, like in the pic?

 

[Simon Bridge]

Hardness I understand, but is the spherical shape necessary?

No.

That would be this article ... under "types of collisions" it says:
Collisions between hard spheres may be nearly elastic, so it is useful to calculate the limiting case of an elastic collision.
It's not well written... it is not saying that only hard-sphere collisions are elastic, but that hard-spheres are useful for modelling collisions.

All you need for elastic collisions is the conservation of the bodies kinetic energy (and momentum).

 

[Danger]

If this is a more practical than theoretical topic, I'll interject another factor that people don't usually consider. This is my observation as a fairly accomplished pool player rather than an educated man. Angular, as well as linear, momentum is transferred to the 2nd sphere. In a game, putting spin on the cue ball changes the angle at which the object ball is deflected or the follow distance if straight. I haven't seen anything that specifically explains how that works, but my suspicion is that there's a masse effect on both balls. (Of course, that aspect relies upon friction with the table surface. It does demonstrate, however, that the object ball picks up opposite spin from the cue ball.) I've taken a couple of championships by knowing that it does happen, even though I'm not sure how.
That same effect wouldn't likely apply to non-spherical objects, as their centres of mass will be out of whack and they might not even be free to rotate at all.

 

[dorker]

All you need for elastic collisions is the conservation of the bodies kinetic energy (and momentum).

So to be clear, can collisions between sufficiently "hard" bodies be assumed as nearly elastic, regardless of shape?

 

[jbriggs444]

So to be clear, can collisions between sufficiently "hard" bodies be assumed as nearly elastic, regardless of shape?

I can think of three caveats.

With two spherical bodies, the impact is always aligned with the centers of both objects. If one neglects surface friction and considers only the normal force then no rotation is imparted to the spheres. So one can ignore rotational kinetic energy. With irregular bodies this is no longer the case. The collision is still elastic, but rotational kinetic energy must be considered.

If the two bodies do not strike "head on" but instead strike a glancing blow then there may be some tangential component to the impact force. If the coefficient of friction is low but non-zero, this may result in some dissipation of energy, regardless of the hardness of the objects. However, hard objects tend to be slippery objects (e.g. steel wheels on steel rails), so this loss will tend to be minimal.

As has been pointed out, it is true as a practical matter that hardness and elasticity tend to go hand in hand. But there is no principle of physics that requires this to be so.

 

[dorker]

^ Ok, thanks for the answers.

 

[Simon Bridge]

Collisions are complicated so it is usually difficult to get a simple answer ;)

The thing is that you keep adding extra bits to your question that also need to be addresed.

1. wether a collision is elastic or inelastic (and to what degree) does not depend on shape.
2. ... also does not depend on "hardness".

How elastic a particular collision is, is determined entirely and only on what happens to the kinetic energy of the bodies involved.

I have an air-track set-up using spring-fenders. The collisions are elastic to the limits of measurement available to the students. Clearly two springs colliding head-on are not "hard" impact surfaces. I have seen a demonstration of electrostatic repulsion between two objects moving on an air table - again, clearly not a hard-surface collision, also very elastic.

@Danger:
In the pool/snooker/billiards situation, the model is complicated by the fact the balls are rolling, and the cue ball is usually not struck (by the cue) through its center of mass. The resulting tangential component is what provides the "spin" on the balls and so the strange effects used in trick shots.
http://www.jimloy.com/billiard/phys.htm

Non-spherical objects also transfer angular momentum in collisions, and you can get much more varied responses since you get, in general, three spin axis. eg. Spin is stable about the longest or the shortest axis but unstable about the middle one: the object "tumbles".
http://www.ph.man.ac.uk/~mikeb/lecture/pc167/rigidbody/stability.html

Just to be clear: any object may not be free to move in some way. You won't find an object that is not free to (for eg) rotate solely by virtue of it's shape. There will be something external restraining it.

 

[dorker]

How elastic a particular collision is, is determined entirely and only on what happens to the kinetic energy of the bodies involved.

So, I'm not sure I get this. Is there any way to know beforehand whether a collision will be elastic? Does it depend on the freedom of movement the bodies will have?

 

[Simon Bridge]

It depends on the mechanical and material properties of the bodies.
There are big thick books listing material properties for many substances, which are determined experimentally.

But for a random collision - you have two arbitrary bodies heading for each other? You can make a good guess - especially if the bodies are simple. Ferinstance, two hardened steel hockey-pucks and the electrostatic example above can be predicted to be very nearly elastic, because we know a lot about the processes involved. Sometimes, however, you just have to bash stuff together. This is why cars have to be crash tested no matter how good the engineer thinks his calculations were.

 

[dorker]

It depends on the mechanical and material properties of the bodies.

Sorry to keep pestering you, but which ones? I'd simply assume it depends on the bodies not getting deformed, but you just said it didn't depend on hardness.

 

[Rap]

It depends on whether the bodies deform without any dissipation of their energy, such as friction, which will convert some of their energy to heat. Hard objects tend to dissipate little of their energy as heat, but, as in Simon Bridge's example, you could have two bodies with springs attached, and if those springs dissipated very little energy as they compressed and expanded during a collision, that would be an elastic collision too, and those objects are certainly not "hard". Its the dissipation of energy that is the key, not the hardness. Two lumps of clay are not hard either, but they dissipate as heat much of the energy that it takes to deform them. If you deform them, they stay deformed, they don't bounce back. Any object which "bounces back" into shape will tend to collide elastically. Hard objects tend to bounce back into shape, but they are not the only objects which have this property. "Hard" rubber balls are certainly not hard like a pool ball or a steel ball, but they collide very elastically.

 

[sophiecentaur]

If a material deforms a lot then there is more chance of energy loss during a collision because there is more 'distance' for the 'force times distance' work done against the lossy forces.

If you take a steel ball bearing and drop it onto a thick steel disc, I'd bet it will bounce a lot higher than the superest of 'superballs' that you can get. It's all relative.

 

[Simon Bridge]

... but a really hard, brittle, ball could chip in some impacts - and a hard shell could dent in some collisions. Suddenly not so elastic. Two steel bells colliding would be less elastic than two steel balls too - more KE lost to sound. The nature of the collision is important too - non-Newtonian fluids exhibit elasticity under sharp blows but not under slow ones.

In general you bet on hard solids being more elastic than soft ones - the simpler the geometry the better that bet.

The details of materials science can get quite complicated: really fat books get written about it. Perhaps if we knew what you wanted to know for that would narrow it down?

To give you an idea - here's a discussion of the material properties relating to deformation:
http://en.wikipedia.org/wiki/Elastic_modulus

... also, briefly, elasticity:
http://en.wikipedia.org/wiki/Elasticity_(physics )

... and brittleness:
http://en.wikipedia.org/wiki/Brittleness