## Billiard balls - where does the energy go?

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tomandlu
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### Billiard balls - where does the energy go?

Sorry, really basic question...

Two billiard balls* roll towards each other at the same speed, and both stop dead when they hit since their vectors are the same length but point in opposite directions. Where does the energy go?

* make this an ideal experiment - i.e. ignore friction, sound, compression
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yurell
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### Re: Billiard balls - where does the energy go?

In an elastic collision (i.e one that conserves both kinetic energy and momentum) they just bounce off each other, travelling back in the direction they came with their initial velocity. If they stop, you have an inelastic collision, and so the energy goes into sound, heat & deformation.

Edit: I should also add: don't apologise. There's no shame in seeking to learn about something you don't understand, and asking questions is a fantastic way to learn.
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tomandlu
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### Re: Billiard balls - where does the energy go?

Thanks - so if this takes place in a vacumn (no sound), and if the balls are 100% undeformable, then all the energy is transformed into heat? (i.e. excitation of the atoms that the ball is made of)
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yurell
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### Re: Billiard balls - where does the energy go?

If it took place in a vacuum and they were undeformable, I'm pretty sure they'd just bounce off each other with their initial speed, with a negligible amount converted to internal energy. That is to say, they won't simply stop when they hit each other.
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douglasm
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### Re: Billiard balls - where does the energy go?

If the balls are 100% undeformable, they would have a nearly perfect elastic collision and bounce off at full speed. The names "elastic" and "inelastic" for collision types are misleading if you think of them in terms of descriptions of the materials involved. Think of it more in terms of the behavior of materials when stretched - a rubber band, which is elastic, will snap back as soon as you let it go; a lump of clay, which is not, will just stay in whatever shape you pulled it into. Rigid undeformable materials tend to have highly elastic collisions because their shapes snap back to normal from the slight deformations involved very quickly, while malleable materials tend to have highly inelastic collisions because they don't snap back into shape at all.

tomandlu
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### Re: Billiard balls - where does the energy go?

Okay, now I'm confused. I've been reading Brian Cox's "Why Does E=mc2" and in the section on vectors, he says the two vectors will cancel each other (i.e. the end-point of each vector will connect to the start-point of the other vector, creating a zero-length vector). In the case of, say, two rubber balls, I can see why they would bounce apart, but I'm obviously misunderstanding something here when it comes to more rigid materials... I wish there was a Newton's Cradle in this house...
How can I think my way out of the problem when the problem is the way I think?

PM 2Ring
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### Re: Billiard balls - where does the energy go?

tomandlu wrote:Okay, now I'm confused. I've been reading Brian Cox's "Why Does E=mc2" and in the section on vectors, he says the two vectors will cancel each other (i.e. the end-point of each vector will connect to the start-point of the other vector, creating a zero-length vector). In the case of, say, two rubber balls, I can see why they would bounce apart, but I'm obviously misunderstanding something here when it comes to more rigid materials... I wish there was a Newton's Cradle in this house...

If the two balls have the same mass and equal but opposite velocity, yes, the vector sum of their momenta will equal zero. This just means that the centre of mass of the system as a whole doesn't move.

It's pretty easy to show that in this scenario of a pair of identical (perfectly elastic) balls hitting each other head-on, the balls' motion is perfectly reversed when they bounce off each other. That's the only possibility that conserves both kinetic energy and momentum. Just play with the algebra, and see it for yourself! Of course, if you get stuck, just give us a yell.

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### Re: Billiard balls - where does the energy go?

tomandlu wrote:Okay, now I'm confused. I've been reading Brian Cox's "Why Does E=mc2" and in the section on vectors, he says the two vectors will cancel each other (i.e. the end-point of each vector will connect to the start-point of the other vector, creating a zero-length vector). In the case of, say, two rubber balls, I can see why they would bounce apart, but I'm obviously misunderstanding something here when it comes to more rigid materials... I wish there was a Newton's Cradle in this house...

Well he's correct for most realistic billiard balls because they make a loud noise when they hit so the energy gets taken away.

If we now move to the world of frictionless vacuums and have perfectly rigid balls, when they collide there is nowhere for the energy to go. It can't go into sound (because there's no air), it can't go into vibration or heat because the ball is "perfectly rigid". The only way to conserve energy in such a system is if the two balls end up moving away at the same speed (but with the direction reversed).

It's also interesting to note that any two-body collision looks like this from the point of view of the system's centre of mass so a Newton's cradle (or at least each individual collision in it) is a perfect demonstration of rigid materials behaving elastically (because the ball coming in moves at speed v and the ball leaving has speed v, from the point of view of the centre of mass both balls are always moving at v/2, they just change direction, first moving together, then away).
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tomandlu
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### Re: Billiard balls - where does the energy go?

Okay, here is my experiment performed under rigerous lab conditions...

http://www.flickr.com/photos/26781520@N08/7843117146/

which I think demonstrates that they stop dead... (but I'm not certain as there may have been one or two minor problems with my precision-engineered equipment)

Edit to add: the more I think about it, the more I think you're right (and that there is something basically wrong with my experiment - shocking, I know) - I'm just having a fundamental problem reconciling zero-length vectors and bouncing apart...
How can I think my way out of the problem when the problem is the way I think?

jaap
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### Re: Billiard balls - where does the energy go?

tomandlu wrote:Okay, here is my experiment performed under rigerous lab conditions...

http://www.flickr.com/photos/26781520@N08/7843117146/

which I think demonstrates that they stop dead... (but I'm not certain as there may have been one or two minor problems with my precision-engineered equipment)

Edit to add: the more I think about it, the more I think you're right (and that there is something basically wrong with my experiment - shocking, I know) - I'm just having a fundamental problem reconciling zero-length vectors and bouncing apart...

Why do you insist on trying to add two unrelated vectors? Just because you can, doesn't mean you should. One represents something (e.g. velocity or momentum) of one ball, the other that of the other ball. How one influences the other depends on what happens in the collision. Generally you can only add vectors if they represent different components of the same property of or acting directly on the same object.

In your experiment, the contents of the colliding cans is messing things up. The contents will continue rotating in the same direction within the cans and cause them to not to reverse direction so easily. They actually start rolling towards each other again after they've bounced apart a bit.
This is very like testing whether an egg is boiled or not. Lay the egg on a hard flat surface, and set it spinning. Momentarily tap it to stop it spinning. A hard boiled egg will stop dead, but a raw egg will start spinning again.

Meteoric
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### Re: Billiard balls - where does the energy go?

When you add together the two momentum vectors, that only tells you the total momentum of the system as a whole. It doesn't necessarily say anything at all about what happens to them individually: after all, the vectors still add to zero if the alignment is off and they don't collide at all. And the vectors still add to zero if they both rebound at full speed.
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tomandlu
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### Re: Billiard balls - where does the energy go?

I'm fairly sure I've now got it -thanks all. With regard to the zero-length vectors (and continuing with the pool-side science), if I'm on a raft and roll one can away from me, the non-zero vector means that I will recoil in the opposite direction. However, if I roll one can forward and one behind, the zero-length vector means that the action will have no effect on the raft.

BTW, as I mentioned, this all came about from reading Why does e=mc2. One (for me) revalation from this book is that C is not defined as "the speed of light" (as in "vegtable" is not defined as "carrot"). Light merely travels at that intrinsic speed because photons are massless (and presumably gravitons, if we ever find them, will also travel at that speed). As a further twist, everything travels at C in spacetime - it's just that light has no time component, so that its speed in spacetime is the same as its speed in 'normal' space. Not news to most on here, I know, but a bit of an Oh me yarm moment for me...
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Gigano
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### Re: Billiard balls - where does the energy go?

Why has Newton's third law not been mentioned yet? "When two bodies interact by exerting force on each other, these forces (termed the action and the reaction) are equal in magnitude, but opposite in direction." --Wiki

So in an ideal system (i.e. no friction with air or ground and a complete elastic collision) the momentum (=mass times velocity) of one ball (or cans) is 100% transferred to the other ball upon collision. So one ball goes with a velocity 5 m/s to the east, another goes with -5 m/s to the west. They meet head on, they interact, their momentums are transferred and they leave the collision with a reverse momentum, i.e. -5 and 5 m/s respectively. Granted the vectors cancel each other out if their initial masses and velocity are equal. But that only tells you the momentum of the entire system has a net value of 0. Each of the balls however has a particular momentum of equal size but opposite direction (e.g. -5 and +5 respectively), so what you see is movement overall. If the net value of the entire system (i.e. 0) would determine the movement of the individual participants, then shouldn't you expect there to be no movement at all to begin with?

The reason why billiard balls, or cans rolling of slopes, do not demonstrate what I described above is because they are not in an ideal system. Billiard bulls are not 100% elastic, more like 99.99...% or thereabout, and there is constant friction with air and the surface on which they roll. The kinetic energy they carry, as mentioned, goes into sound, heat and movement of air particles which they collide with as they go.
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tomandlu
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### Re: Billiard balls - where does the energy go?

Gigano wrote:Why has Newton's third law not been mentioned yet? "When two bodies interact by exerting force on each other, these forces (termed the action and the reaction) are equal in magnitude, but opposite in direction." --Wiki

Indeed - I'd allowed myself to get sidetracked from this by an essentially unrelated aspect of motion. BTW, does the third law give a solution to the unstoppable/immovable objects paradox, or do the infinities involved break the maths?
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Gigano
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### Re: Billiard balls - where does the energy go?

The infinities do not really screw up the maths, but they make sure the situation can never happen. An unmovable object would have to have infinite inertia, and thus infinite mass. There is no infinite mass in the universe, therefore an unmovable object cannot exist. Similarly, an unstoppable force would require infinite energy. There is no infinite energy in the universe, therefore an unstoppable force cannot exist.

So the third law isn't required to solve the paradox from a pragmatical viewpoint. The premises that are required to allow for the paradox to exist at all, are just not possible in our universe. But a more philosophical approach is to suppose a universe which does allow unstoppable forces exists. Such a universe would not allow unmovable objects, else the unstoppable force is not unstoppable. Conversely, a universe which allows unmovable objects cannot allow unstoppable forces.

All in all, in no universe will an unstoppable force ever encounter an unmovable objects. Logica dixit.
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Sandor
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### Re: Billiard balls - where does the energy go?

Billiard balls also have (to some extent) the problem of spin, similar to the cans experiment, at least if you roll them towards each other. Newton's cradles and Clackers (or Clacker Balls, or Ker-Bangers) don't spin, so are better tools to experiment with. Try a YouTube search for clackers.

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### Re: Billiard balls - where does the energy go?

Gigano wrote:All in all, in no universe will an unstoppable force ever encounter an unmovable objects. Logica dixit.

Depending on your definition of unstoppable or immovable. If we simply contend that no finite force can move an immovable object and no finite force can stop the unstoppable force then we merely require that they both be infinite in value in which case, by allowing hyperreal values for physical quantities, we can solve the problem. Granted, the unstoppable force would not be truly unstoppable or the immovable object truly immovable and you'd have to already know "how" infinite the force and mass were, there is a way to formulate the problem such that it is meaningful.
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douglasm
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### Re: Billiard balls - where does the energy go?

My definition, an unstoppable force and an immovable object cannot both exist simultaneously in the same universe. If either does exist, then the definition of the other is impossible to satisfy.

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### Re: Billiard balls - where does the energy go?

As I said, this is certainly the case for strictly immovable objects and unstoppable forces. I was saying that, if we only require them to immovable or unstoppable against all finite forces (rather than all forces) then the two can coexist.
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### Re: Billiard balls - where does the energy go?

Sandor wrote:Billiard balls also have (to some extent) the problem of spin, similar to the cans experiment, at least if you roll them towards each other.

This is worth calling out. When you're dealing with billiard balls you have to take into account where in the ball the contact takes place, that's how you put spin on the ball after all. Ignoring friction, if you have two balls hit each other head-on in an elastic collision you'll arrive at the situation where the balls are still spinning like they're trying to go one way, but are actually moving in the opposite direction. Scratching out the math (warning: I could have screwed this up) suggests the kinetic energy of rotation and translation would be equal solid cylinder (like in the video), for a solid sphere the former is 40% of the latter. The video ofcorse is complicated by the fact that the stuff in the cans is probably fluid.

Soralin
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### Re: Billiard balls - where does the energy go?

Sandor wrote:Billiard balls also have (to some extent) the problem of spin, similar to the cans experiment, at least if you roll them towards each other. Newton's cradles and Clackers (or Clacker Balls, or Ker-Bangers) don't spin, so are better tools to experiment with. Try a YouTube search for clackers.

Or Newton's cradle: http://www.youtube.com/watch?v=jid7Nlzfet8

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### Re: Billiard balls - where does the energy go?

tomandlu wrote:Okay, here is my experiment performed under rigerous lab conditions...

http://www.flickr.com/photos/26781520@N08/7843117146/

which I think demonstrates that they stop dead... (but I'm not certain as there may have been one or two minor problems with my precision-engineered equipment)

Did you notice how they bounced apart and then actually moved back towards each other?

This is because they have liquid in the middle. You can use this effect to tell whether an egg is raw or cooked http://www.youtube.com/watch?v=aPswkvEmQ7c
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tomandlu
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### Re: Billiard balls - where does the energy go?

Sandor wrote:Billiard balls also have (to some extent) the problem of spin, similar to the cans experiment, at least if you roll them towards each other. Newton's cradles and Clackers (or Clacker Balls, or Ker-Bangers) don't spin, so are better tools to experiment with. Try a YouTube search for clackers.

Also worth mentioning that players will often take advantage of this behaviour by choosing how high or low they strike the ball with the cue, thus controlling its behaviour after it has hit the second ball.
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### Re: Billiard balls - where does the energy go?

I think I had the correct answer to this scenario but for completely the wrong reason. My though was that when the balls collide they would *exchange* their energy and so the ball moving left would then move right and vice versa giving the appearance of a perfect bounce.
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tomandlu
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### Re: Billiard balls - where does the energy go?

Aelfyre wrote:I think I had the correct answer to this scenario but for completely the wrong reason. My though was that when the balls collide they would *exchange* their energy and so the ball moving left would then move right and vice versa giving the appearance of a perfect bounce.

Isn't that what they do? Or rather, can it be stated one way or the other, and both statements be correct? I've no idea, but I'm thinking... very, very slowly...
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eSOANEM
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### Re: Billiard balls - where does the energy go?

Not quite because that explanation would require that the energy they impart has a direction inherent in it whereas, because energy is scalar, it does not.

It really is because of both energy and momentum that the system behaves exactly as it does with each being exactly reversed and trying to solve the system using just one will either introduce odd/wrong assumptions or sneak the other one in there somewhere.
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