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Perfectly centered break of a perfectly aligned pool ball rack
- aspensmonster 13y agoI didn't realize there was both mathoverflow.net and math.stackexchange.com. A quick glance at both looks like they serve largely the same purpose. Cool modelling in any case!
- verandaguy 13y agoI believe MathOverflow - a formerly non-SE site - targets graduate-level questions, while math.SE has a broader target user base.
- jc4p 13y agoYeah, MathOverflow was a SE 1.0 site (when we had white-label Stack Exchange networks rather than internally made ones) which has since been added into the real network with the SE 2.0 rollout. Math.SE and MathOverflow seem to function well together even they though have similar areas of interests.
- verandaguy 13y agoCould you explain the expression, "white-label?" First time I've heard it.
- jeroen 13y agoThis gives a decent explanation: http://en.wikipedia.org/wiki/White-label_product http://en.wikipedia.org/wiki/White-label_product
- jc4p 13y agoThe Wikipedia article explains the concept, but basically the initial business plan for Stack Exchange was to sell the software to companies / groups that wanted to run their own system versus the all in-house management we have today.
- neltnerb 13y agoThis is beautiful. I'd be curious about the results of a similar monte carlo simulation (or random variable statistics if possible) that had identical force laws but introduced random gaps between the balls.
- kungfooguru 13y agoWeird, wasn't expecting to find that it was by my brother in law!
- jimbelk 13y agoHi! This seems like a neat website -- kind of like early reddit.
- girvo 13y agoThat's the aim, to be honest :)
- kungfooguru 13y agoWoo! It is sort of a slashdot/reddit, since it focuses (almost) only on tech and startups.
- lbarrow 13y agoJim Belk was my math professor in college -- we always knew he was a much better programmer than he let on!
- stephancoral 13y agoIt's pretty neat seeing someone from Bard on HN
- jimbelk 13y agoHi!
- chubot 13y agoWhat does the pattern of motion look like once they start bouncing off walls? What about if the table is frictionless?
- hmsimha 13y agoI'm no physics expert, but I can account for the second one -- if there's no friction, the balls just keep rolling
- Someone 13y agoNot necessarily; it might be possible to have some balls collide in such a way that at least one comes to a stop. At least one ball will keep moving though, and it isn't clear to me whether it is possible to prevent that moving ball from, eventually, hitting that stopped ball again. I think that would require at least three balls (ignoring pockets)
- zodiac 13y ago> it might be possible to have some balls collide in such a way that at least one comes to a stop. can you give an example?
- eru 13y agoIf you just have two identical balls.
- deleted 13y ago[deleted]
- hmsimha 13y agoI can't see how friction is factoring into the directions the balls initially scatter, as it's not taking the rotation of the balls into account. With that being said, wouldn't the balls all go in the same initial direction as they do in the linked example? Aka, the ones that are moving would move in that direction. Theoretically they could eventually bounce off walls and collide into each other in such a way that they eventually stop (though then there's the possibility of other balls that are still moving colliding into them later).
- TheLoneWolfling 13y agoThis doesn't take into account rotation though :( See http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/index.html http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/in... for including rotation.
- joosters 13y agoAlso doesn't take into account the direction of the nap of the cloth on the table - balls rolling 'up' will travel differently from those rolling 'down'. Would this also mean that the speed of the balls has to be taken into account? The differing effects of friction on the surface would surely alter as the balls move & spin slower?
- nostromo 13y agoWhat about air friction? And the the Coriolis effect? And does observing the balls cause their wave function to collapse? :P
- joosters 13y agoYes! We demand to know whether the pool table was in the northern or southern hemisphere :)
- 7483263789 13y agoTut tut! You're forgetting that the earth is an oblate spheroid, and gravity exerts greater downward force at the poles, than near the equator. We also need to know the precise latitude of the pool table!
- kaivi 13y agoOne might as well consider the gravitational gradient exerted by the surrounding terrain and significant celestial bodies, as well as the balls themselves. Should we collaborate on assembling such simulation, taking in account all known physical laws? Or has it been done already? Imagine how cool it would be to toggle different forces or constants and see the result change.
- rcthompson 13y agoCan anyone explain the "rules" for a 3-way simultaneous colision? Freshman physics only covered collisions between 2 objects, and I've never figured out how to generalize it.
- waqf 13y agoRight, with 3 balls you can't just do an impulse calculation like you can with 2 (too many unknowns, too few equations). The author is simulating the collision as an event which happens gradually over an (admittedly short) period of time, using a differential equation, instead of something that happens all at once. That's why he has to talk about the elasticity model he's using, F ∝ (2-d)^1.5, and how the results vary when you change it.
- tel 13y agoCould you get more equations by symmetry?
- waqf 13y agoYes … in the symmetric case. That might give you enough to solve the first one-against-two collision. But the subsequent collisions in the pool break problem won't be symmetric (the velocity of the "incoming" ball is not perpendicular to the axis joining the centers of the other two).
- rcthompson 13y agoHmm. How does the elasticity model provide the additional constraint needed to give an unique solution?
- waqf 13y agoIt provides a way to apportion the impulse between the two simultaneous collisions. It behaves as a differential equation because the instantaneous force in each collision depends (via the elasticity model) on the instantaneous velocities of the balls, which in turn depend on how much momentum has already been transferred.
- taliesinb 13y agoReally cool problem! And great animations. I was disappointed to find that the associated notebook (http://math.bard.edu/belk/code/BilliardsHertz.nb http://math.bard.edu/belk/code/BilliardsHertz.nb) doesn't work correctly in either Mathematica 9 or 10 -- I think the author used 8. It seems like NDSolve falls down: NDSolve::smpf: Failure to project onto the discontinuity surface when computing Filippov continuation at time 0.`. >> and the solution shows the cue bool passing through all the others..
- sillysaurus2 13y agoThere's a Mathematica 10 now, already? Interesting. I'm still a happy 8 user, myself. Is there much reason to upgrade? It'd probably be good for them to pay more attention to backwards compatibility...
- ballstothewalls 13y agoI think he is confused. Wolfram's site only has Mathematica 9 listed
- prakashk 13y ago> I think he is confused. You don't mean that the GP (taliesinb) is confused, do you?. He works for Wolfram Research, developing Mathematica.
- ballstothewalls 13y agoThat is what I meant, but I guess I look like the fool now. I still don't see that Mathematica 10 has been released, but I guess it is close.
- nswanberg 13y agoIt hasn't been released, but you can effectively get it on a Raspberry Pi: http://wolfram.com/raspi http://wolfram.com/raspi
- IgorPartola 13y agoSomewhat meta, but at what point does Stack Overflow transition from being a useful, time-saving tool and become a waste of everyone's time? The top answer author clearly put a ton of time on the answer. We are all now reading this. Is anyone able to apply any of the knowledge they gained from this Q&A to anything productive?
- nardi 13y agoYou need to stop and smell the billiard-ball-shaped roses, my friend. Not every pursuit needs be in the name of industry. Some people enjoy mathematical puzzles for their own sake.
- dredmorbius 13y agoFor those whose day job is modelling pool breaks, this could be a time-saving tool. Actually, for anyone whose work involves modeling physics and collisions in general. The fact that it describes a phenomenon with which a great many people are familiar and at least casually interested is only one aspect of this. And there's a long history of useful discoveries and advances coming from gaming and gambling (going back to Blaise Pascal).
- srean 13y agoI presume you know the story of Feynman and spinning plates [1]. Sense of play is important for productivity, sometimes it is more than just important, it becomes a necessity [1] http://www.physics.ohio-state.edu/~kilcup/262/feynman.html http://www.physics.ohio-state.edu/~kilcup/262/feynman.html The diagrams and the whole business that I got the Nobel Prize for came from that piddling around with the wobbling plate. -- Richard Feynman
- jimbelk 13y agoThis is a good question, and I sometimes do put too much time into answering questions on Math Stack exchange. Right now, though, I'm fairly happy with the time I put into this simulation. First of all, there's the fact that I did this on a Friday evening, after I came home from work. (I spent about six hours on it, and maybe another hour or two on Saturday morning). There are probably more productive uses of my time, but it was fun, and the most likely alternative was probably something like "read reddit" as opposed to "do something productive". It's nice to have leisure-time activities that are at least marginally related to my career. Speaking of which, here's a few reasons that this was helpful for me: 1. I got a little better at programming in Mathematica. I use Mathematica a lot for both my teaching and research, so it always helps to practice. 2. I learned about a whole field of physics called contact mechanics, which I otherwise wouldn't have been aware of. It's part of my job as a math professor to know about random things like this. 3. I got some experience with modeling in applied mechanics problems. This may be helpful for me in the future -- as a professor at Bard, I regularly have to supervise senior math majors who want to work on research-level problems. The next time a senior who's interested in applied mechanics wants a research problem, I'll have one available. That's how it's helpful for me, but I suppose it's less clear that this was helpful for the rest of the world. I guess, at the very least, it certainly seems to have entertained a bunch of people, and maybe someone learned something about modeling or differential equations or physics.
- Glyptodon 13y ago... but if you hit it hard enough what's the maximum number that will go in?
- jimbelk 13y agoAssuming there's no friction on the table, the chances ought to be pretty high that they'll all eventually go in.
- JetSpiegel 13y agoThat's why I always open pool games with a full-strength shot. Now there's SCIENCE to prove it.
- chinpokomon 13y agoOnce you start figuring out the elastic collisions with the rails, there's even more to consider. Rails are positioned such that the balls strike them on their northern hemispheres. Because the rails are a wedge, the ball will get pinched between the rail and the table. To understand this collision, you need to take into consideration the rotational momentum of the ball. This is easier to solve when it is a completely orthogonal collision with no lateral rotations. If the ball strikes the rail on an angle, the rails twist to absorb the impact and will recoil with a force that isn't just a simple reflection, but that also has a force vector tangent to the ball and back towards the origin. This directly affects the spin of the ball when it leaves the rail. Combined, a bounce off the rail is a very complicated collision. The net result of which has balls that strike the rail at lower velocities closely matching a reflection, but if struck with a higher velocity, the reflection straightens out into a bounce more perpendicular to the rail. In short, the rails are highly dependent upon the velocities at impact, and perhaps more so than at break, are affected by rotational momentum.
- manglav 13y agoWhile this is great for a theoretical model, most professionals also try to put forward spin on the cue ball, to offset the collision and break apart the rack further. You may have noticed small white burn marks from the heat generated by the cue ball spinning on the felt, from the location where most people break.
- a_soncodi 13y agoThis is mostly incorrect. The chosen spin depends on several factors including the breaking position (as in breaking from "the box") and ball count. In this case it appears to be Eight Ball. The burn marks are caused by the downward force due to an inclined cue. Similar marks occur when performing a legal jump. This is why many tournaments provide a cloth scrap for breaking. Furthermore, the cue ball may hop to the rack and impact the head ball above its equator, causing the cue ball to jump in the air, reducing the likelihood of being kicked into a pocket. Top spin or forward roll is not required for this. Source: I play pool semi-professionally.
- manglav 13y agoAgree on the spin, I was assuming a perfectly vertical break. Abou the burn marks, see here http://www.pooldawg.com/article/pooldawg-library/table-maintenance http://www.pooldawg.com/article/pooldawg-library/table-maint... http://www.cuesight.com/types-of-pool-table-felt.html http://www.cuesight.com/types-of-pool-table-felt.html Yeah, I hate it when the ball pops up, but my opponent looks cool catching it, heh. I also play pool. Best game for lounging I feel.