newtonian mechanics - What determines whether a pool ball will bouce backwards after colliding with another pool ball?

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?]

Answer

Backspin!

Those shots in which the cue ball "draws" backwards after hitting the target ball involve backspin. Without backspin, the cue ball cannot reverse direction.

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!