Using a rigid body pool physics simulator that is restricted to the 2D plane it is easy to demonstrate that if you keep increasing the force the cue imparts on the cue ball all the balls will eventually go in every time.
Now imagine that I want to make as many balls in off the break as possible in real life. I have two cues available to me, a 1oz and a 50oz. I lift weights and can wield the 50oz cue. Which do I use?
If you are vastly strong it doesn’t matter. All that matters is what is the cueball’s speed after it moves away from the cue towards the racked balls. Faster is better.
The whole question is what best imparts speed to the cueball.
Baseball has gone back and forth on heavy bats vs. light. Any given batter can always get more bat speed with a lighter bat. But the lighter, more flexible, bat may not be as effective at transferring that extra speed to the ball. And ball speed is what delivers distance. Bat speed is just a means towards ball speed.
Similar effects apply to a cue. When the cue impacts the ball the ball speeds up and the cue slows down. Meanwhile your muscles are trying to push the cue, then cue+ball, at a constant speed. Or even better, at a constantly increasing speed.
Assuming the cues are equally rigid simplifies the discussion a bunch. If you are strong enough to absorb the cue/cueball impact with no slowing of cue motion, the weight is immaterial. If not, then the heaviest cue you can accelerate to full speed within your stroke length is best. Any cue heavier than that will not strike the cueball with your personal maximum possible speed.
And of course, the stronger you are, the faster (probably) you can accelerate any given cue weight prior to impact. Issues with overtraining and muscle-boundness could get involved here.
Last of all, I assumed the cues were equally rigid regardless of weight. That may fail to be true for the lightest weights. And loss of rigidity further reduces transmission efficiency.
If you’re real wimpy you could be in a situation where you need a real light cue to get any cue speed, but by the time they’re that light, they’re also increasingly non-rigid. In which case you’d find yourself in the same situation as MLB players. With a very real tradeoff between cue speed and cue transmission efficiency. And where the answer to your personal optimization question may differ from some other guy who’s slightly stronger or weaker, faster or slower.
Late Add:
I’m thinking the OP was looking for a theoretical treatment more than practical pool-playing advice.
Ref Tripolar’s article.
His physics is good, but he’s operating solely within the region where you don’t have adequate arm strength to accelerate the cue to top speed. He’s in the region where lighter cue => faster arm => faster cue => faster ball. This is where the transfer efficiency benefits of the heavier cue are overshadowed by the raw speed gains of the lighter one.
I think LSL covered the physics pretty well.
When I played with my own cue, I used a 17oz. Aluminum. Straight as an arrow, and slid well. Lot’s of folks thought it was wimpy (it was pink, so that may have had something to do with it 
)
I’m a big guy with long arms. I could really get some speed behind that cue, and had no problem with making an impressive break.
This seems obviously false. Friction will eventually stop all the balls. If, on the other hand, you are simulating on a perfect frictionless table with perfect angle of incidence = angle of rebound cushions, there is some chance a ball will get into a cyclic bounce which just retraces its path.
OT: Is there such a thing as a (documented) record for most balls sunk off a break?
I think LSLGuy covered most of it, but I’ll add a couple of things.
Assuming the collision between stick and ball is close to elastic, the maximum energy transfer happens when they weigh the same. Like a Newton’s cradle, when the stick hits the ball, it will come to a halt and the ball will gain all of the energy and momentum. Cue balls weigh 5.5 oz, so in this sense you want a 5.5 oz stick.
The problem is that a lightweight stick is hard to put a lot of energy into. There some minimum where you can’t accelerate your arm any faster. And again, very roughly speaking, the maximum energy transfer between your arm and the stick is when they weigh about the same. A human arm weighs maybe 8 lbs, so you’d want your stick to weigh that much to put maximum energy into it.
So in practice you want your stick to be somewhere in the middle. I’m going to make a wild guess and say that the geometric average is where the theoretical peak is. For 5.5 oz and 8 lbs, that comes to a 27 oz stick, which isn’t so far off from real stick weights. Of course there are some problems with my calculation, like the fact that your entire arm isn’t swinging at full velocity, just the lower part of it, which probably means that the real number is even closer to typical stick weights.
In short, I’d say that you can’t know the optimal weight without experimentation, but theory does suggest that weights in the 20 oz range are already in the ballpark of being optimal, and you wouldn’t want to go a factor of 2 above or below this value.
@enipla: I was old school and had a wooden 21oz. Worked for me, scrawny as I was. Then again, I played 14.1, not nine-ball or some other philistine thing. So even on the break it was a finesse game, not slop.
Lighter cues gave me less fine speed control & so less control over where the cue ball stopped…
Suppose you had a cue going 40 mph with no hand attached to it and another cue going 40 mph with a hand still pushing it. How much difference would it make. How much speed is left in the cure after it makes contact with the ball.
The trick is speed the cue is going at the time it hits the cue ball…
-With that said, throw a tennis ball with your hand and see how fast you can throw it.
-Next throw a bowling ball with your hand and see how fast you can throw it.
(You can get a lighter object/cue going a faster speed easier than a heavier object!)
I feel more comfortable putting some force behind a heavier stick when I break. I can feel how it’s moving better that way and keep it under control. So the physics of a machine using a heavier or lighter stick to break are simpler than the physics for a human which have to include the ability of the nervous system to control and compensate the movement of the stick.
After performing the necessary calculations I have come to the following conclusion: I use a house stick to break because I don’t want to break mine. It’s going to be a heavier wooden stick. So heavier is better.
This is a really difficult question to answer due to the difference in table quality.
If you’re playing on a bar quality table where the pockets are a designed for the player to be able to make more shots so the games ‘turn over’ faster to benefit the proprietor you’re going to get a different result than if you’re playing on a good quality ‘tournament’ quality table where pockets are regulation size. Shooting too hard on a tournament table will cause shots to “wiggle” and not fall in the pockets that are not close to dead center. This of course will throw your calculation way off. I don’t shoot any harder than necessary and make some shots that ordinarily would have “danced” in the pocket for me had I shot it harder as I wasn’t as accurate as I could have been. Shooting easy sometimes is beneficial.
Phu Cat
Yes, but a bowling ball is very heavy. Even a heavy pool cue is relatively light; it’s more like throwing two tennis balls than a bowling ball.
Kinetic energy =1/2 mass*velocity squared, right? Doesn’t that mean a heavier cue at a certain speed will impart more energy to the cue ball compared to a lighter cue at the same speed?
The probability of a ball on a frictionless table getting into such a cycle is vanishingly small, though. From a given starting point on the table, such a cycle will have a finite number of bounces off the top/bottom walls, and a (possibly different) number of bounces of the left/right walls. There’s a limit to the number of top/bottom and left/right bounces that a closed cycle can have; otherwise, they’ll need to be so tightly spaced that one of the bounces will coincide with a pocket. Since a ball can start off going in an infinite number of directions (between 0° and 360°), there are an infinite number of trajectories from each starting point; and so the probability of a pool ball on a frictionless table entering into a closed cycle (assuming all trajectories are equally likely) is zero.
As far as adding friction goes, one could in principle hit the balls harder to make them roll further before coming to a stop. Of course, this might require so much force from the cue and so much speed from the balls that you would have to worry about breaking the cue, shattering the balls, setting the felt on fire due to friction (careful with those massé shots!), nuclear fusion, etc.
I am not following this. In this situation you are not looking for the cue stick to have 0 energy after the collision (i.e., 100% transfer); you are looking to maximize the gross amount of energy transferred to the ball. Increasing either the weight or the speed of the stick, or both, will increase the amount of energy transferred to ball, even though the stick will still have some energy after the collision.
The dynamics behind this are very complex. The cue is part of a system that includes the shooter’s arm. Muscles act like springs, different people have different strengths and arms weights. Technique plays a big part, with some people much better at transferring arm/body momentum to the cue. For example, Nick Varner has a very powerful break, and he’s a tiny guy who weighs something like 115 lbs. The optimum cue weight will be slightly different for each person as a result of all this.
Given all the complexities and difficulty modeling the entire system, the best way to figure out optimum cue weight is through experiment and data collection. That’s exactly what Bob Byrne and Bob Jewett did - they removed the foot rail from a pool table set up in a garage, then started shooting balls off the table and marking where they hit the ground. From there they were able to work out the velocity of the cue ball as it left the table.
They tried multiple cue weights between 9.5 and 25.9 ounces, with multiple people, all experienced pool players. After 67 shots in total, they collected the data and graphed it.
What they found was that first, there was a lot of shot-to-shot variance. These guys were all college level players, and presumably had pretty good technique but their velocities varied by as much as 20%. So the sensitivity of the test was pretty low. However, they found that the optimum range of cue weights clustered around 18 to 21 ounces. That’s the range that produced the most consistently fast cue balls for the players as a group.
Within that range, the player’s own variance tended to dominate. However, one clear result is that there’s very little advantage to a heavy cue - not a single player was able to hit the ball as fast with a 25.9oz cue as they could with a cue between 18 and 21 ounces.
The speed difference in that range were really small, though. For example, one player’s 3 shots were measured at 23.72 mph with a 20-ounce cue, 23.29 mph with a 19.5 ounce cue, and 22.85 mph with a 17.9 ounce cue. Notice that the lightest and heaviest cues had better speeds than the middle one, meaning that the differences really just came down to the player’s technique from shot to shot.
The conclusion was that you might as well just use the cue you are comfortable with, so long as it’s in that range between 18 and 21 ounces. Personally, I have a 19oz playing cue, and I use it to break as well. That’s what Nick Varner does too. If you play with a really whippy cue, you might want a stiffer one for breaking, but don’t sweat the weight. Just use whatever feels right.
Right, but for a moving stick with a given amount of energy, then maximizing the transfer maximizes the energy that ends up in the ball.
The trick of course is getting that particular amount of energy into the stick in the first place. And that’s a coupling between your arm and the stick, and is at the most efficient when the stick weighs the same as the effective swinging mass of your arm.
So there’s some balance between these two–getting the most energy into the stick demands a heavy stick, but delivering that energy efficiently into the ball demands a light one. Somewhere between the two is the optimal point. Any heavier than this, and you do worse because the stick-ball efficiency goes down faster than the arm-stick efficiency goes up; while any lighter than this and the reverse is true.
If you’re still curious, let me know and I can run some basic numbers for you to demonstrate the concept.
@Sam Stone: Excellent post. Real data and everything. Bravo. Thank you.
One issue those experimenters had with taking such little data is that everybody who took a shot or three had grown up playing with 16 to 21oz cues. Or maaaybe 22 oz tops. I’ve not seen/heard of heavier being in actual use anywhere. So any cue outside that normal weight range would be a new and unfamiliar experience.
It’d be interesting to have each player knock 50 or 100 practice balls off the end using heavier cues each day for a week. Then come back the next day and gather your data. Now you’ll have players with recent practice up to 26oz or whatever.