Your goal is to sink all the striped or solid balls, and finally the 8 ball. You're assigned stripes or solids based on the first ball sunk (after the opening shot). But watch out, if you sink the 8 ball too early, you lose the game!

When a player has potted all of their (solid or striped) balls, they must pot the black 8 ball to win the game. Caution: if you pot the 8 ball BEFORE your other balls, you automatically lose. Fouling when shooting for the 8 ball does NOT result in a game loss, except if you pot BOTH the cue ball and 8 ball with your shot.

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Many people don't know what exactly gives billiard balls that unmistakable feel. Over the centuries, billiard balls have been made from a wide range of materials, including wood, ivory, celluloid, a proprietary plastic material known as Bakelite, and others.

These materials are incredibly durable, and they're capable of being formed into highly precise products that meet exacting standards. Phenolic resin in particular offers a remarkably smooth, shiny, long-lasting billiard ball with a one-of-a-kind surface that makes a successful break feel like heaven.

There are many great billiard ball brands out there, but if you're looking for the best of the best, you can never go wrong with Brunswick Billiards or Aramith. These two brands offer some of the highest-quality ball sets on the market. While premium brands can come with more of an investment up front, they are built to last.

If you want to optimize your pool table experience, choosing a top-quality set of billiard balls is a great place to start. Two of the most important factors to consider when you're searching for your perfect set are roundness and material.

To your eye, virtually every set of billiard balls is going to look perfectly round. However, even the slightest discrepancies can make a significant difference in the way the balls play. Make sure the set you purchase has been tested multiple times to ensure a perfect round shape with no defects.

I'm no knowledgeable pool player, but I've noticed that sometimes when the cue ball hits another pool ball, they roll together; and sometimes the cue ball bounces back. And I have a very, very rough sense that a hard, sharp, and even strike of the cue ball tends to make it bounce back more while a slower or more angled strike will make it roll forward after collision. Can anyone give a more rigorous analysis of the phenomena, or point me to a resource for this? I've tried googling but haven't see anything that really seems to address this as far as I can tell.

[Edit: Upon more contemplation, I suppose a more general question is: In a collision, what determines how much of the combined momentum of the system is distributed to the parts? So in cars colliding, or pool balls, or a skater on ice throwing a baseball--what features of the system determine the amount of momentum imparted to each component?]

Consider what happens when the cue ball is not spinning at all when it hits the target ball. The cue ball will come to a dead stop if it hits the target ball straight on. Think of Newton's cradle. The cue ball will continue moving forward (but at an angle) if a non-spinning cue ball hits the target ball obliquely.

The cue ball always moves forward after striking the target ball if the cue ball is rolling without slipping whilst hitting the target ball. A rolling cue ball will initially stop if it hits the target ball straight on. The cue ball will still be spinning, however, and this spin will soon make the cue ball start moving forward again. When a rolling cue ball hits the target ball obliquely, the collision will change the cue ball's direction and the spin will accentuate the forward motion.

The only way to combat these effects is to have the cue ball spinning backwards when it strikes the target ball. A backspinning cue ball that hits a target ball straight on will initially stop, but now the backspin will make the cue ball reverse direction.

So how can one make the cue ball have backspin? The answer is simple: Strike the ball below center. How much below depends on the distance to the target ball. This is easy if the target ball is close to the cue ball: Strike the cue ball a bit below center. You'll need to strike the cue ball a bit further below center if the target ball is further away. When the target ball is very far away (across the length of the table), it's very hard to have the cue ball spinning backwards at the point of collision.

You need to take care in your shot and how far from off-center you hit the cue ball. Hit the cue ball too far off-center and you'll hear a nasty "clink" sound. You've just miscued; the cue ball won't move anything like you planned. And maybe you've even ripped the table, bad move!

So in the center-of-mass frame, only the angles change! In particular, a straight collision of two idealized balls of the same mass has a clear outcome: if we describe the situation from the viewpoint of the table, the cue ball stops and transmits the whole energy to the previously static ball that was hit. This ball we hit is moving with the same speed that previously belonged to the cue ball.

If you hit it roughly, it's likely that the ball will be moving at some speed but rotate at a lower speed than what is needed for it to roll on the table without friction. I am not saying that the ball is actually rotating backwards. Instead, it is not rotating at all, so it will never try to revert the direction of motion, as you observed.

But if the cue ball isn't sufficiently rotating forward, it will be more likely to bounce back. The reason is simple. As the cue ball A hits the previously static ball B, the ball B starts to move forward and it also starts to roll forward, because of the friction between B and the table. But by momentum conservation, B has to act on A that starts to roll backwards (the point at which the two balls are meeting is moving up which has the "opposite" implications for the two balls). This backward rotation of the hard cue ball A will tend to send A in the backward motion, too.

The initial speed of the cue ball is immaterial -- slowing down the cue ball is the same as slowing down time. The force constant $10^{11}$ has no real effect as long as it's large enough, although it does change the speed at which the initial collision takes place.

For this model, the entire collision takes place in the first 0.2 milliseconds, and none of the balls overlap by more than 0.025% of their radius during the collision. (These figures are model dependent -- real billiard balls may collide faster or slower than this.)

The following animation shows the forces between the balls during the collision, with the force proportional to the area of each yellow circle. Note that the balls themselves hardly move at all during the collision, although they do accelerate quite a bit.

After the collision, some of the balls are travelling considerably faster than others. The following table shows the magnitude and direction of the velocity of each ball, where $0^\circ$ indicates straight up.

For comparison, remember that the initial speed of the cue ball was 10 units/sec. Thus, balls 11 and 15 (the back corner balls) shoot out at more than half the speed of the original cue ball, whereas ball 5 slowly rolls upwards at less than 2% of the speed of the original cue ball.

Glenn the Udderboat points out that "stiff" balls might be best approximated by a force response involving a higher power of the distance (although this isn't the usual definition of "stiffness"). Unfortunately, the calculation time in Mathematica becomes longer when the power is increased, presumably because it needs to use a smaller time step to be sufficiently accurate.

As you might expect, most of the energy in this case is transferred very quickly at the beginning of the collision. Almost all of the energy has moves to the back corner balls in the first 0.02 milliseconds. Here is an animation of the forces:

While the simplicity of this behavior is appealing, I would guess that "real" billard balls do not have such a force response. Of the models listed here, the intial Hertz-based model is probably the most accurate. Qualitatively, it certainly seems the closest to an "actual" break.

It might be interesting to compare the solutions of this equation with suitable boundary conditions to simulations along the line of Jim Belk's for a very large pool ball rack (large enough at least so that one can follow the shock wave for an appreciable length of time). 0852c4b9a8

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