making the break

So, the Spousal Unit took off this morning for a conference somewhere in Wisconsin and left the Resident Feline and I alone with the brand new flat-screen TV. This is what happens when I ask the Spousal Unit to stop off at Circuit City on the way home from the office because I need a more advanced science-y calculator. Not that we're complaining, because the new TV is teh awesome! We played hooky from calculus, plopped ourselves on the couch and wasted the afternoon watching Witchblade on DVD. Anyone else remember that short-lived series on TNT ("We know drama!"), loosely based on the graphic novel series published by Top Cow?

Witchblade was one of my guilty pleasures -- guilty because, frankly, it was a very uneven production, with tacky symbolic imagery, major chewing of the scenery by the supporting cast, and some truly horrific dialogue at times. (There's an entire scene in the first non-pilot episode, "Parallax," where the characters literally speak in koans. While playing chess. It's cringe-inducing.) But the series also had a killer soundtrack, a genuinely compelling underlying "mythology," and Yancy Butler starring as Sara "Pez" Pezzini, a NYC cop who finds herself wielding a mysterious ancient bracelet that turns out to be pretty damn useful in a fight.

Butler made the series, frankly. She took a comic book character known more for her exaggerated pulchritude and skimpy outfits, and transformed her into a street-smart, tough, sexy, emotionally complex woman -- who just happened to play a mean game of pool in the bargain. Yancy Butler kicked butt, literally and figuratively.

My favorite scene  in the two-hour pilot is Pez taking on every guy in the local bar in successive games of pool, and handily sinking every shot after the break in each game. She pockets a nice chunk of change, too. This, frankly, is a common fantasy among women. I am no exception: in my dreams, I can walk into any bar and wow the locals with my prowess.

Alas, far from being a skilled pool shark, I am utterly inept at the game. I'm not being modest. It's a thrill if I manage to hit the cue ball correctly, and if it also hits one of the object balls and gets it to move a tad, huzzah! Actually sinking one of the object balls pretty much makes my week.

Lots of people throughout the centuries have had a similar fascination with some form of pool, notably billiards. (For simplicity's sake, I'm not going to go into the many variations made popular all over the world. Follow the various links and you'll learn more than you ever wanted to know about cue-stick games.) The game has its roots in a lawn game resembling croquet, dating to 15th century Europe. Perhaps folks tired of having their games rained out or something, because eventually the game evolved into an indoor tabletop version, whereby balls were shoved (not struck) with wooden sticks called maces. Originally there were only two balls, as well as a wicket (hoop) and a stick as a target, but eventually people figured out that you really just needed the balls and cue sticks and a few pockets around the table to have a kick-ass game. There's even a reference to billiards in Shakespeare's Antony and Cleopatra.

The iconic image of pool or billiards (in the U.S., at least) is the 1961 movie The Hustler, starring Paul Newman. It's a dark, fairly gritty film, actually, but for some reason it inspired a billiards revival, even though pool was a game of ill-repute in many American communities in the 20th century. The game went highbrow again two decades later, when Newman played an aging pool shark mentoring Tom Cruise's ambitious young hustler in 1986's The Color of Money. And while the prevailing image is one of a boozy boy's club, women have always indulged in billiards, although they weren't officially organized until 1976, with the birth of the Women's Professional Billiards Association. Just a few years before, a grandmother named Dorothy Wise won five U.S. Open tournaments, proving once and for all that it wasn't just a "man's game."

There's a certain degree of practiced skill involved, even to become adept at the basics, even more so if one aspires to learn some of the more advanced shots, or tricks. And like most sports, there's a great deal of physics involved in the seemingly simple game of pool, as evidenced by the large number of online resources outlining the specifics in detail. It's standard classical Newtonian stuff, mostly: overcoming the cue ball's inertia, accounting for friction from the table's green felt surface, the transfer of momentum between the cue ball and the object ball when they collide (it's not a perfectly elastic collision, but close enough), and so forth.

The paths the balls take after colliding depends on the above factors, as well as the angle at which the cue ball hits -- which in turn depends on where the cue stick hits the cue ball, which depends solely on the player's skill and control (or the lack thereof, in my case). Draw and Follow shots, for example, involve (respectively) hitting the ball below center to put a backwards rotation on it, or hitting above the center to put a forward spin on it. If we can figure out how to measure the mass, position and velocity of each ball on the table at the time of collision, we should in principle be able to predict the path and outcome of the shot.

Ah, but that's just too easy for some people. I found this entertaining online tutorial via Google on Quantum Billiards: what might it be like to play pool at the subatomic level, with balls the size of protons? Things can change in an instant when an observation is made, you can't now both the position and momentum of any ball at the same time, and each event has many possible outcomes, not just one. You're pretty much just taking shots in the dark. And don't forget about quantum tunneling! Normally a bank shot lacks sufficient energy to hop over (or through) the cushioned barrier of the billiard table; instead, it is reflected off at predictable angles. Not so if the ball is the size of a proton. Because its tiny mass creates large uncertainties, there's a much higher probability it could go right through the cushioned barrier. Electrons do it all the time, why not subatomic billiard balls?

Of course, if you really want to make things interesting, you need a spherical cow model for billiards, and a recent paper accepted by Physical Review Letters apparently offers just that. Physicists at Boston University studied what would happen during the initial "break shot" of a billiards game in an ideal setting: namely, with no dissipation of energy (I assume this means a perfectly elastic collision, with nothing lost to heat, noise, etc.) and an infinite billiard table. Heck, if we can have billiard balls the size of protons, why not infinite tables? (Or even quantum versions of Cruise and Newman?)

Basically, they created an ideal gas and then sent the particles careening all over the place, from a central starting point. Their conclusion: "Just as in real billiards, progressively more particles become mobile as the collision cascade develops." But there was an interesting twist. The initial break is, naturally, asymmetric, with various balls flying off in different directions and speeds. But in the idealized model, as the balls (or particles) expanded outward, the region became nearly spherically symmetric around the initial point of collision. In fact, it looked for all the world like a shock wave generated from an explosion. Now that is freaky.

Shock waves do form when the speed of a gas changes by more than the speed of sound. Wherever this happens, according to Wikipedia, "sound waves traveling against the flow reach a point where they cannot travel any further upstream and the pressure progressively builds in that region, and a high pressure shock wave rapidly forms." Something similar happens with supersonic jets: parts of the air around the plan travel at exactly the speed of sound, along with the aircraft, but the plane leaves a pile-up of these sound waves in its wake. The waves are forced together and compressed -- sort of an amplification effect -- ultimately merging into a shock wave that spreads out sideways.

Thunder is a naturally occurring sonic boom, and yet another example of a shock wave. And of course, explosions generate shock waves, such as when a bomb goes off. It just hadn't occurred to me that colliding billiard balls might also produce a shock-wave phenomenon. But when the collisions are viewed in slow motion, as in the YouTube video below, it does seem a bit more explosively violent than when observed at full speed:

Here's one last bit of trivia to relieve the Monday morning doldrums. Apparently the cracking sound of a bullwhip is a tiny sonic boom. The end of the whip has far less mass than the handle, so swinging the whip sharply, energy is transferred down the length of the whip. The velocity of the whip increases as mass decreases, such that ultimately the end (called the "cracker") moves faster than the speed of sound -- one of the first human inventions to break the sound barrier. I'll bet Sara Pezzini swings a mean bullwhip, when she isn't shooting pool.

Comments

[Calvin]

Nice discussion on pool! Being something of a fan(atic) myself, I was pretty stoked to see something mentioned on this blog related to pool (haven't gone through all the postings yet so I may have missed something.) To be a reasonably good pool player one must have at least a minimal understanding of the physics involved in the game, particularly in the way the balls interact with each other, the rails, and the cloth. Jim Loy's site does a great job of discussing these things, but the second site is a little harder to take seriously (every reasonably experienced pool player knows that the six ball is GREEN, for instance.) It has some impressive equations (which are WAY over my head as an armchair physicist) but some things seem to be missing in the ball collision example. One page describes the angle of deflection as being roughly 90 degrees (the tangent line at the point of impact.) While this is true at the instant of contact, it rarely remains that way for very long. The examples don't speak to the angular momentum of the cue ball (the draw, follow, or side spin you mention in your post) or the effects of throw (the effect that cue ball side spin has on an object ball at the moment of impact.) These things cause not only a change in the path of the cue ball but also in the path of the object ball as well.

I've been a long time armchair physicist, which is to say I find the subject of physics fascinating and study it as it relates to other interests, but have no formal education in it beyond the one high school class I took. For instance, as a windsurfer, it's good to know the physics of wind and water. For the longest time I would wipe out thinking that the board's fin had ripped off completely when it actually hadn't. (Imagine suddenly finding yourself sliding sideways across the water at 20MPH praying that you don't catch the leading edge. It does loosen the sphincter somewhat.) It was fascinating to discover the reason it occurs is related to the reason why military submariners don't want to spin the propellers on their subs too fast (cavitation is the scientific term.)

All that being said, I really enjoy reading your blog posts and learning about various other aspects of math/science that I wouldn't normally go research on my own.

Calvin.

[Glenn N]

Ah, who cares about the science? I wanted to hear more about Witchblade! I too have been watching it lately and am happy to discover that I missed many of the first season when it aired.

I really appreciate your blog as an educational tool for my friends (I have advanced degrees in geophysics, so I benefit less, but still enjoy it).

[Jennifer Ouellette]

There's this bracelet, see, with mystical powers.... :) Actually, there IS a bit of temporal physics involved, i.e., when Sara undergoes the Periculum (a bizarre test of worthiness) and is told that while we think of time as linear, it's really more like individual frames in a film. It's our perception that makes it appear to be linear and continuous, when in fact one moment can touch another moment ("frame") at any time. I think brian Greene said something similar in THE FABRIC OF THE COSMOS, without all the mystical trappings. :)

And I think we could probably come up with a hypothetical type of material out of which the Witchblade might have been made. "A branch from the Tree of Good and Evil" is just a bit too vague.... Ferromagnetic liquids/solids might be a contender...

At least I gave you a saucy shot of Sara Pezzini!

[Porlock Hussein Junior]

A deterministic spherical-cow analysis of billiards is fine, and so is quantum billiards, but you've omitted one other fashionble analysis. Chaos theory needs to be represented:

"Even the motion of a frictionless billiard ball becomes completely unpredictable after only 11 collisions, owing to the uncertainty principle's limit on describing its initial state."

This is from a really splendid obituary for Edward N. Lorenz, by Kerry Emanuel, Science 23 May 2008: Vol. 320. no. 5879, p. 1025;
http://www.sciencemag.org/cgi/content/full/320/5879/1025

Can't speak for anyone else, but I've been waiting for years (not being a physicist and exposed to this sort of thing in journals) to see exactly this merger of chaos and uncertainty.