Reverse Throw

RicNic said:

Published papers on this concept appear to be in need of correction if they say that there is none or very little effect on the cue ball's motion.

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Exactly. Some of the published books and papers are so far off in this ignoring of the counter effect it's laughable. And then people will tell you they can hold a cueball dead still and throw the object ball some amazing sideways distance to, say, get a stop shot position on a break out shot in 14.1.

It's a shame.

Freddie

RicNic said:

Does anyone know how much reverse throw is applied to the cue ball when throw is applied to the object ball?

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As others have pointed out, the effect is equal and opposite. The resulting effect on CB speed and angle are different depending on the type of shot. For example, the effect on stun and rolling-ball shots are described here:

For cut shots with English, the CB will pick up a little tangent line speed with outside English greater than the gearing amount, and will lose a little tangent line speed with inside English or outside English less than the gearing amount.

"Reverse throw" is also useful to help "hold" the cue ball. For more info, see:

Regards,
Dave

RicNic said:

......We would all benefit if someone would explore this effect in some mathematical way, and applied in a practical way.

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When the balls are close enough to each other and/or you're hitting hard enough such that the cueball doesn't lose any significant backspin on the way to the object ball (or gain more topspin), there is a method of determining the cueball's direction once it reaches natural roll after the collision. I call it the Bottom-Center-Arrow method, or B-C-A for short, in that it's easy to remember.

Imagine a circle centered on the ghostball with the bottom of the circle running through the center of the cueball. This circle represents the face of the cueball from the shooter's perspective. To determine the CB's roll direction after the collision for any vertical offset (no sidespin applied), draw a line from the center of the real cueball parallel to the line of centers between the ghostball and the object ball. This will intersect the tangent line at 90 degrees, call it point A. Thus, we have a triangle with the CB at vertex B (bottom of the circle), the ghostball at C (center of the circle) and point A from which we'll draw an arrow such that it intersects the vertical axis of the large circle. This yields the CB's direction once roll sets in, given that vertical tip offset on the face of cueball. Here's a diagram:



The relevant point here is that friction, amongst other things, has an effect on this idealized geometry. Below is plot of the deviations from this ideal due to CB "throw" at various vertical offsets and cut angles.



(Note:the discontinuities in some of the curves are transition points from partial to full sliding during the collision.)

As you can see, the largest departures happen at close to stun at smallish cut angles. But these deviations can be essentially eliminated by applying an amount of inside english determined as follows. Swing the vector going from (3/5)R below center to the desired vertical tip offset, parallel to the direction of the line of centers between the ghostball-object ball. The location of the tip of this vector yields the amount of inside english to use (as seen on the large circle epesenting the face of the CB) in order to negate the effects of friction. A couple of diagrams should help, one for draw, one for follow:





I should note that the above correction assumes that a certain component of the CB's spin (i.e., along the ghostball-object ball line of centers) is unaffected by the collision. While this is certainly not exactly true, I believe it's true enough compared to the effect of the friction on the CBs other spin component along the tangent line. Also, the use of the inside english will generate another deviation: post-impact swerve.

There are other deviations due to mismatched ball weights and the less than perfect elasticity of the collision, but I'll leave it at that for now, since we're discussing CB throw.

Jim

RicNic said:

Does anyone know how much reverse throw is applied to the cue ball when throw is applied to the object ball?

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Depends on what kinda reverse throw you use really.What kinda cb to.And you need to find out if the object ball is the kind that will accept reverse throw.

Im confused ..

Jim,

Excellent post. I plan to look into this more closely when I can find some time. I'll also quote it on one of my resource pages at some point.

... remainder of the message deleted until I find time to think through the results ...

Thanks,
Dave

Jal said:

When the balls are close enough to each other and/or you're hitting hard enough such that the cueball doesn't lose any significant backspin on the way to the object ball (or gain more topspin), there is a method of determining the cueball's direction once it reaches natural roll after the collision. I call it the Bottom-Center-Arrow method, or B-C-A for short, in that it's easy to remember.

Imagine a circle centered on the ghostball with the bottom of the circle running through the center of the cueball. This circle represents the face of the cueball from the shooter's perspective. To determine the CB's roll direction after the collision for any vertical offset (no sidespin applied), draw a line from the center of the real cueball parallel to the line of centers between the ghostball and the object ball. This will intersect the tangent line at 90 degrees, call it point A. Thus, we have a triangle with the CB at vertex B (bottom of the circle), the ghostball at C (center of the circle) and point A from which we'll draw an arrow such that it intersects the vertical axis of the large circle. This yields the CB's direction once roll sets in, given that vertical tip offset on the face of cueball. Here's a diagram:

View attachment 216279

The relevant point here is that friction, amongst other things, has an effect on this idealized geometry. Below is plot of the deviations from this ideal due to CB "throw" at various vertical offsets and cut angles.

View attachment 216278

(Note:the discontinuities in some of the curves are transition points from partial to full sliding during the collision.)

As you can see, the largest departures happen at close to stun at smallish cut angles. But these deviations can be essentially eliminated by applying an amount of inside english determined as follows. Swing the vector going from (3/5)R below center to the desired vertical tip offset, parallel to the direction of the line of centers between the ghostball-object ball. The location of the tip of this vector yields the amount of inside english to use (as seen on the large circle epesenting the face of the CB) in order to negate the effects of friction. A couple of diagrams should help, one for draw, one for follow:

View attachment 216280

View attachment 216281

I should note that the above correction assumes that a certain component of the CB's spin (i.e., along the ghostball-object ball line of centers) is unaffected by the collision. While this is certainly not exactly true, I believe it's true enough compared to the effect of the friction on the CBs other spin component along the tangent line. Also, the use of the inside english will generate another deviation: post-impact swerve.

There are other deviations due to mismatched ball weights and the less than perfect elasticity of the collision, but I'll leave it at that for now, since we're discussing CB throw.

Jim

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dr_dave said:

Jim,

Excellent post. I plan to look into this more closely when I can find some time. I'll also quote it on one of my resource pages at some point.

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I really appreciate the kind comment Dr. Dave. (I realize at this point you haven't verified my results). I do hope you come up with the same conclusions!

Yes, I did mean inside english. It doesn't cancel the throw and will increase it in some cases, as you well know. But it balances its effects on the various components that go into the CB's final roll direction....according to my math of which I'm fairly certain. But, of course, it would be great if you get the time to confirm it (or otherwise!) and add your insights.

Note that the method suggests one unnecessary correction: when using pure stun. While harmless in a way, this is the only vertical offset (0) in which the method doesn't really apply.

Thanks again,
Jim

You know it's a good thread when...

...dr_dave, Jal, Bob Jewett, and Cornerman all post in it.

jsp said:

...dr_dave, Jal, Bob Jewett, and Cornerman all post in it.

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You forgot Jsp???

Jal said:

Swing the vector going from (3/5)R below center to the desired vertical tip offset, parallel to the direction of the line of centers between the ghostball-object ball. The location of the tip of this vector yields the amount of inside english to use (as seen on the large circle epesenting the face of the CB) in order to negate the effects of friction. A couple of diagrams should help, one for draw, one for follow:

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That's really interesting. Can you post your math for this as well? Based on your diagrams I believe I can ascertain the formula which leads to your conclusion, but I am having trouble quantifying the CB spin based on the tip location. I don't believe the relationship between tip position and CB spin is entirely linear, due to the inelastic nature of the tip and CB collision. In addition I think many of the factors in the tip/CB collision would make quantifying this spin extraordinarily difficult, such as tip end mass, tip shape, and tip frictional co-efficient.

Even if we assume a perfectly elastic collision for the stroke, and a linear relationship between tip position and CB spin, it seems in your diagrams that CB speed is not a factor either. This is definitely a departure from the standard thinking on throw, unless I have been dreadfully misled

If your math checks out, it will definitely change my game. I was expecting the line depicting the tip position for zero throw to be parabolic (or perhaps circular), returning to the vertical centre at 3R/5 above the horizontal centre. Clearly that kind of thinking would have me adjust in the wrong direction for follow cut shots.

OnTheMF said:

That's really interesting. Can you post your math for this as well?

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I don't have the math for the tip offset graphical interpretation, but the foundation physics and math along with a graphical interpretation for the rolling CB case can be found here:
See Equation 24 on page 4 and Equation 39 on page 6. Also see the explanation and illustration on page 7 based on Bob Jewett's July '08 Billiards Digest article (see page 13 here).

Maybe Jim can post a summary for us for the general case. I also plan to work through and eventually post something for this (with credit to Jim). Jim has made a nice contribution here by extending the graphical approach to all draw and follow shots (not just roll shots).

OnTheMF said:

Based on your diagrams I believe I can ascertain the formula which leads to your conclusion, but I am having trouble quantifying the CB spin based on the tip location. I don't believe the relationship between tip position and CB spin is entirely linear, due to the inelastic nature of the tip and CB collision. In addition I think many of the factors in the tip/CB collision would make quantifying this spin extraordinarily difficult, such as tip end mass, tip shape, and tip frictional co-efficient.

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In the ideal case, the spin-to-tip-offset relationship is definitely linear. See:

And taking into account cue and ball mass, and tip inefficiency, the relationship is still linear. See:

OnTheMF said:

Even if we assume a perfectly elastic collision for the stroke, and a linear relationship between tip position and CB spin, it seems in your diagrams that CB speed is not a factor either.

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The final CB trajectory angle depends only on the spin-to-speed ratio (AKA spin-rate factor, percentage English, "tips" of English). For more info and illustrations, see:

OnTheMF said:

This is definitely a departure from the standard thinking on throw, unless I have been dreadfully misled

If your math checks out, it will definitely change my game. I was expecting the line depicting the tip position for zero throw to be parabolic (or perhaps circular), returning to the vertical centre at 3R/5 above the horizontal centre. Clearly that kind of thinking would have me adjust in the wrong direction for follow cut shots.

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Object ball throw is a different question. Zero throw occurs at the "gearing" amount of English; and you are correct ... this does vary nonlinearly with cut angle. For details, see:

Diagram 2 in my January '07 BD article shows how the amount of English required for "gearing" varies with cut angle. The relationship is nearly linear over a fairly wide range of cut angles typical in good pool play. A 1/2-ball hit requires 40% English.

Regards,
Dave

OnTheMF said:

That's really interesting. Can you post your math for this as well? Based on your diagrams I believe I can ascertain the formula which leads to your conclusion, but I am having trouble quantifying the CB spin based on the tip location. I don't believe the relationship between tip position and CB spin is entirely linear, due to the inelastic nature of the tip and CB collision. In addition I think many of the factors in the tip/CB collision would make quantifying this spin extraordinarily difficult, such as tip end mass, tip shape, and tip frictional co-efficient.

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OnTheMF, thanks for your interest. My apology, but I really don't have the time at the moment to go into all the math. Dr. Dave's material has/can likely answer most or all of your questions. Since he's looking into this, I don't know, but he may eventually put up one of his technical proofs on the subject.

OnTheMF said:

Even if we assume a perfectly elastic collision for the stroke, and a linear relationship between tip position and CB spin, it seems in your diagrams that CB speed is not a factor either. This is definitely a departure from the standard thinking on throw, unless I have been dreadfully misled

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As it happens, the indicated correction (using the vector from the (3/5)R below center offset) works regardless of the actual amount of throw. So it is CB speed/surface condition independent. You could always add more vertical offset to compensate for the frictional loss of spin during the collision, but that would depend on CB speed and cut angle (i.e., coefficient of friction). The inside english adjustment doesn't.

OnTheMF said:

If your math checks out, it will definitely change my game.

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But which way?

OnTheMF said:

I was expecting the line depicting the tip position for zero throw to be parabolic (or perhaps circular), returning to the vertical centre at 3R/5 above the horizontal centre. Clearly that kind of thinking would have me adjust in the wrong direction for follow cut shots.

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I have to confess, I'm not completely following this.

I'll truly try to answer any questions, supplying the math as necessary (in addition to Dr. Dave's) shortly. Very briefly, the CB's velocity at roll after the collision is derived from the general relation:

Vroll = (5/7)[Vi - (2/5)WXRb]

where Vi is its initial velocity, W its spin, and Rb the vector from the center of a uniform sphere to the point of contact with the surface. This can be derived in various ways, and it sounds like you might already be familiar with it.

The CB's spin about the horizontal axis (X R) is given by:

RWx =-5/2(Bz/R)V

where Bz is the vertical offset, R the ball's radius, and V its speed.

If we set the y-axis (unit vector j) along the cueball to ghostball direction, with x-axis to the right (unit vector i) for a cut to the left, then using the above relations, the cueball's velocity after the collision and natural roll sets in is:

Vroll = (5/7)[Vsin(C)cos(C)i + (Vsin^2(C) + (Bz/R)V)j]

where V is its pre-impact speed in the j direction and C the cut angle.

However, it's better to express the unit vectors i and j in terms of a unit vector along the tangent line t and one along the object ball-ghostball line of centers n:

i = tcos(C) - nsin(C)
j = tsin(C) + ncos(C)

Substituting these in the expression for Vroll and a little algebra, you get (I hope!):

Vroll = (5/7)(V/R)[Rsin(C)t + Bzj]

(Note that j reappears in this final expression by combining terms.)

Thus, if you scale V to a ball's radius, the CB's final roll direction is the vector sum of Rsin(C) along the tangent line and the vertical tip offset in the cueball-ghostball direction.

Sorry, but I have to run. Hope that illuminates something.


Jim

Jal said:

When the balls are close enough to each other and/or you're hitting hard enough such that the cueball doesn't lose any significant backspin on the way to the object ball (or gain more topspin), there is a method of determining the cueball's direction once it reaches natural roll after the collision. I call it the Bottom-Center-Arrow method, or B-C-A for short, in that it's easy to remember.

Click to expand...

Jim, are you familiar with the geometric constructions in Coriolis' 1835 book? If not, I think you'd enjoy them. (Unfortunately, the current English translation is rife with misprints on almost every page and diagram, but you're well-versed enough in the physics to get past those. Or you could try the original French that has fewer printing errors).

Your B-C-A is very similar to his carom construction with arbitrary friction (including sliding vs what he calls "adherence" contacts), where he also shows that the greatest deviations are near stun with fuller hits. I enjoy these kinds of geometric complements to algebraic derivations that help reinforce good intuitions, and I've created java applet 'live' versions of all of his diagrams so you can play with the parameters (I'll put them up online when I have the time.) I'll probably create one for yours as well sometime soon.

I like how you account for the 5:2 relationship due to moment of inertia by normalizing the initial velocity to a ball radius so you can use the tip offset directly as an aiming aid. Coriolis instead chose to normalize it to a ball width so he could reference the OB contact point (one way to sidestep the swerve issue), but the approach is the same. Both let you add linear and angular velocity vectors to determine the final rolling direction. He didn't directly mention the inside spin compensation in the text, but it's clear from his drawing the amount of inside required to rotate the contact friction impulse vector in the direction that cancels the deviation. I like the directness of your using the CB tip position for this.

Coriolis preferred drawing to scale including the parabolic curving portion of the ball path, so he extended the idea with a line that represents table friction and gravitational acceleration (f*g) that he used in order to determine the point that the curving stops and straight-line rolling begins. (Ah, the pre-calculator/computer days of analog geometric calculation lol.). Btw, Coriolis also included the compensations for ball mass inequality and collision inefficiency that you mentioned were possible to extend the diagram to show.

Unfortunately, the main thing that Coriolis missed -- or should I say 'dismissed'? -- in his analysis is the significance of the velocity-dependence of friction during collisions. As Bob Jewett, Dr. Dave, and others have shown, that's kind of a big deal Maybe if Coriolis was around pocket games instead of carom games he would've paid more attention.

Robert

Robert Raiford said:

Jim, are you familiar with the geometric constructions in Coriolis' 1835 book? If not, I think you'd enjoy them. (Unfortunately, the current English translation is rife with misprints on almost every page and diagram, but you're well-versed enough in the physics to get past those. Or you could try the original French that has fewer printing errors).

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Until recently, financial restraints have prohibited me from purchasing much in the way of pool related materials, so I haven't had the chance. One of these days I'll break down (maybe) and start delving into his work, as well as Wayland Marlow's. Dr. Dave, along with Ron Shepard and Bob Jewett, by themselves provide a wealth of billiard physics in a style that I'm more at home with. But as for the original French, after two years of it way back when, unfortunately all I can remember is Je ne parle pas Francais.

Robert Raiford said:

Your B-C-A is very similar to his carom construction with arbitrary friction (including sliding vs what he calls "adherence" contacts), where he also shows that the greatest deviations are near stun with fuller hits. I enjoy these kinds of geometric complements to algebraic derivations that help reinforce good intuitions, and I've created java applet 'live' versions of all of his diagrams so you can play with the parameters (I'll put them up online when I have the time.) I'll probably create one for yours as well sometime soon.

Click to expand...

I sure hope you do make them available and let us know. You must have been very busy!

I should have known that "my" B-C-A method was preempted by that Coriolis guy - didn't he have better things to do? Actually, years ago someone posted a method on this forum which was very similar, or more likely, exactly the same. But I wouldn't know what to search for to find it. Bob J. used the same approach when the cueball is rolling before impact, if memory serves.

Robert Raiford said:

I like how you account for the 5:2 relationship due to moment of inertia by normalizing the initial velocity to a ball radius so you can use the tip offset directly as an aiming aid. Coriolis instead chose to normalize it to a ball width so he could reference the OB contact point (one way to sidestep the swerve issue), but the approach is the same. Both let you add linear and angular velocity vectors to determine the final rolling direction. He didn't directly mention the inside spin compensation in the text, but it's clear from his drawing the amount of inside required to rotate the contact friction impulse vector in the direction that cancels the deviation. I like the directness of your using the CB tip position for this.

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Thanks Robert for the generous remarks....but again that Coriolis fellow? Didn't he know when to quit! I see that you immediately understood the core of the adjustment (changing the direction of the friction force), but that's hardly surprising.

Robert Raiford said:

Coriolis preferred drawing to scale including the parabolic curving portion of the ball path, so he extended the idea with a line that represents table friction and gravitational acceleration (f*g) that he used in order to determine the point that the curving stops and straight-line rolling begins. (Ah, the pre-calculator/computer days of analog geometric calculation lol.). Btw, Coriolis also included the compensations for ball mass inequality and collision inefficiency that you mentioned were possible to extend the diagram to show.

Click to expand...

Well hell, I mean c'mon... Seriously, I hope these are included in your Java displays. I know that if the cueball happens to be a little lighter than the object ball, it will tend to compensate for the inelasticity over a limited range of cut angles and speeds. But is there a way of formulating a more general approach?

Robert Raiford said:

Unfortunately, the main thing that Coriolis missed -- or should I say 'dismissed'? -- in his analysis is the significance of the velocity-dependence of friction during collisions. As Bob Jewett, Dr. Dave, and others have shown, that's kind of a big deal Maybe if Coriolis was around pocket games instead of carom games he would've paid more attention.

Click to expand...

Ah Ha, I knew he was a hack...though I'm still awestruck by his construction for masse shots. It's not too hard to derive if you already know the answer, but what could have possibly led him to the answer?

Jim

Bob Jewett said:

Fortunately, you don't have to know "la plume de ma tante" from "la petite bourgeoisie" since Dave Nadler translated it into English. See http://www.coriolisbilliards.com/ for details.

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Merci, Bob. (It's all coming back.)

Jal said:

I haven't had the chance. One of these days I'll break down (maybe) and start delving into his work, as well as Wayland Marlow's.

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If you eventually do dive into Coriolis, let me know. I have many corrections for the English edition I can share that I haven't sent to Nadler (the translator) yet. The English version is incredibly valuable, but there are tons of printing errors, including faithful reproduction of several from the original French Coriolis does make a few actual mistakes in the work (I guess he's human after all), but they don't significantly affect his results.

Marlow's work would've been more interesting if he hadn't fixated on the notion that the 'proper' cushion height is 0.7R. He based his entire cushion analysis on that idea even though no tables I've ever seen have them at that height, and the work suffers as a result. (Table manufacturers have empirically evolved the current height over many years for a reason - maybe he should've thought a bit about why that is?)

I sure hope you do make them available and let us know. You must have been very busy!

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I created the interactive diagrams quite a while ago during my second complete reading of the book. They certainly clarified some things for me that were hard to imagine completely from a few static figures (e.g. the dynamic transition from adherence to sliding and its effects on construction 'behavior'). They were originally for my own use (although Ira Lee has used them), so they're unpolished and a bit cryptic for those unfamiliar with the work. I meant to put them online once I made time to write some explanatory text, but I got distracted by other things. However, now that you've got me thinking about it again, I'll try to get something up in a couple weeks after I get back in town.

I should have known that "my" B-C-A method was preempted by that Coriolis guy - didn't he have better things to do?

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You mean like introducing the ideas of "work" and "kinetic energy" that we use throughout physics today? Don't sweat it - billiard scientists and others have been rediscovering his methods/results for years

Well hell, I mean c'mon... Seriously, I hope these are included in your Java displays. I know that if the cueball happens to be a little lighter than the object ball, it will tend to compensate for the inelasticity over a limited range of cut angles and speeds. But is there a way of formulating a more general approach?

Click to expand...

Here's how Coriolis might have modified your construction geometrically to try to account for them: Your point A always lies on a circle that passes through points B and C with a diameter that represents the initial velocity vector. Assume the CB retains a fixed percentage alpha of its initial velocity along the impact line of centers due to an inefficient collision and/or it being a heavier ball (assume the difference is sufficiently small to ignore nonlinear terms). That adds a alpha*V*sin(cutAngle) vector to the CB's tangent line and spin vectors to get the new initial and final angles. Add this geometrically by using a modified ghost ball location C' that's found by moving C closer to B so the modified length BC' equals (1-alpha)*BC. (A lighter CB would modify it by expanding the GB circle instead of shrinking it.)

I've attached a diagram similar to yours that shows this for NR and RNR tip heights for a 39 degree cut with an exaggerated alpha of 0.15 to make it easier to see. There are dashed lines for the ideal case and solid vectors for the modified case including the new stun line.

Ah Ha, I knew he was a hack...though I'm still awestruck by his construction for masse shots. It's not too hard to derive if you already know the answer, but what could have possibly led him to the answer?

Jim

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The masse construction is brilliant, but I think if you check out the rest of the book you'll agree it follows rather logically from his graphical approach in general. (Btw, two oft-ignored details of the masse construction are that it assumes immediate tip-ball separation so it can be treated as two impacts, tip-ball and ball-table, and it assumes sliding between ball and table. If the impacts aren't seperable or there is adherence, it needs to be modified or ignored altogether. It's a great reference, though.)

Robert

Robert Raiford said:

If you eventually do dive into Coriolis, let me know. I have many corrections for the English edition I can share that I haven't sent to Nadler (the translator) yet. The English version is incredibly valuable, but there are tons of printing errors, including faithful reproduction of several from the original French Coriolis does make a few actual mistakes in the work (I guess he's human after all), but they don't significantly affect his results.

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Would you be willing to post the errors and corrections here? I would certainly like to mark them in my copy of the book. I actually reviewed the original draft for David Nadler and found many errors (both in Coriolis' original work and in the translation). Apparently, and not surprisingly, I also missed some. Having written several books, I appreciate how difficult it is to have a large work be completely error-free, especially when there are multiple people (authors, editors, reviewers, typesetters, illustrators, etc.) involved in a project.

Again, please post your corrections so we can benefit from them. I'm sure David Nadler would also appreciate the info (e.g., for future printings).

Regards,
Dave

Robert Raiford said:

If you eventually do dive into Coriolis, let me know. I have many corrections for the English edition I can share that I haven't sent to Nadler (the translator) yet. The English version is incredibly valuable, but there are tons of printing errors, including faithful reproduction of several from the original French Coriolis does make a few actual mistakes in the work (I guess he's human after all), but they don't significantly affect his results.

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I appreciate the offer and if we're both around here if/when I get a copy, I would very much like to have your corrections. That would obviously make it a much easier read. But, per Dr. Dave, if it isn't too daunting a task, perhaps you could make them available here?

Robert Raiford said:

Marlow's work would've been more interesting if he hadn't fixated on the notion that the 'proper' cushion height is 0.7R. He based his entire cushion analysis on that idea even though no tables I've ever seen have them at that height, and the work suffers as a result. (Table manufacturers have empirically evolved the current height over many years for a reason - maybe he should've thought a bit about why that is?)

Click to expand...

Thanks for the heads up, although I'll still be interested to see what he had to offer.

Robert Raiford said:

I created the interactive diagrams quite a while ago during my second complete reading of the book. They certainly clarified some things for me that were hard to imagine completely from a few static figures (e.g. the dynamic transition from adherence to sliding and its effects on construction 'behavior'). They were originally for my own use (although Ira Lee has used them), so they're unpolished and a bit cryptic for those unfamiliar with the work. I meant to put them online once I made time to write some explanatory text, but I got distracted by other things. However, now that you've got me thinking about it again, I'll try to get something up in a couple weeks after I get back in town.

Click to expand...

That would be great if you get the time.

Robert Raiford said:

Here's how Coriolis might have modified your construction geometrically to try to account for them: Your point A always lies on a circle that passes through points B and C with a diameter that represents the initial velocity vector. Assume the CB retains a fixed percentage alpha of its initial velocity along the impact line of centers due to an inefficient collision and/or it being a heavier ball (assume the difference is sufficiently small to ignore nonlinear terms). That adds a alpha*V*sin(cutAngle) vector to the CB's tangent line and spin vectors to get the new initial and final angles. Add this geometrically by using a modified ghost ball location C' that's found by moving C closer to B so the modified length BC' equals (1-alpha)*BC. (A lighter CB would modify it by expanding the GB circle instead of shrinking it.)

I've attached a diagram similar to yours that shows this for NR and RNR tip heights for a 39 degree cut with an exaggerated alpha of 0.15 to make it easier to see. There are dashed lines for the ideal case and solid vectors for the modified case including the new stun line.

Click to expand...

I see the logic and agree with the modification. But the "sin(cut angle)" threw me off for a while - shouldn't that be cos(cut angle)? One difficulty in easily accounting for inelasticity is that the coefficient of restitution (I realize your "alpha" isn't the COR, directly) varies, I believe, with the CB speed component along the impact line. Thus, so will alpha. I'm basing this on tests done by Cushioncrawler (Cue Chalk Board). I have them on another semi-defunct computer, but I believe I remember that the greater the speed, the smaller is the COR. This, of course, makes it cut angle dependent as well.

Robert Raiford said:

The masse construction is brilliant, but I think if you check out the rest of the book you'll agree it follows rather logically from his graphical approach in general. (Btw, two oft-ignored details of the masse construction are that it assumes immediate tip-ball separation so it can be treated as two impacts, tip-ball and ball-table, and it assumes sliding between ball and table. If the impacts aren't seperable or there is adherence, it needs to be modified or ignored altogether. It's a great reference, though.)

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Dr. Dave has described the departures from the predictions as well. I think a significant one is squirt. Although it may be relatively small in the inclined plane of the cue, it's projection onto the horizontal plane magnifies it considerably at the steep inclinations used with masse shots.

Jim

dr_dave said:

Would you be willing to post the errors and corrections here? I would certainly like to mark them in my copy of the book. I actually reviewed the original draft for David Nadler and found many errors (both in Coriolis' original work and in the translation). Apparently, and not surprisingly, I also missed some. Having written several books, I appreciate how difficult it is to have a large work be completely error-free, especially when there are multiple people (authors, editors, reviewers, typesetters, illustrators, etc.) involved in a project.

Again, please post your corrections so we can benefit from them. I'm sure David Nadler would also appreciate the info (e.g., for future printings).

Regards,
Dave

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I briefly corresponded with David Nadler in 2007. Mostly, it was me thanking him for all of that tedious hard work he put into the translation, but I did try to be productive and include a few corrections that he was nice enough to include in the errata pdf available from his site for the book.

Unfortunately, he didn't seem very interested in diving into the details of corrections based on the content, such as the obviously mistakenly drawn location of point L in figure 24 of Coriolis' original that he reproduced when recreating the figures for the book. I never got a reply to my last email, so I didn't pursue it further. I greatly appreciate the tremendous effort he put into it, and the last thing I want to do is bug the guy with anything that may be taken as criticism as opposed to being helpful.

However, I don't mind bugging you - especially, when you asked so nicely Here's a .doc file I created at the time with some nicely formatted errata through p.65 (about halfway through Ch.2):

RaifordCoriolisErrata.doc

As you can see, there's 7 pages worth of corrections just for the beginning alone. I was going to go through the entire book like that and send it to him, but I wasn't as sure as you are that it would be received so positively after his lack of response. The rest of the corrections are still only penciled in the margins of my copies of the English and French texts. I hope to eventually pretty them up and put them online for the benefit of other readers, but it'll have to wait until I have more free time. Now that I know there are at least 2 interested parties, that may even actually happen

Robert

UPDATE: I just viewed the errata doc file in LibreOffice/OpenOffice, and the equations aren't reproduced well. I suggest using MS Word to view it, if possible.

RicNic said:

Thanks to all for considering this question. As a number of you have suggested, there is an equal and opposite reaction on the cue ball, the magnitude and direction, of which, depend upon the specific conditions, i.e., stun, draw, etc. Published papers on this concept appear to be in need of correction if they say that there is none or very little effect on the cue ball's motion. We would all benefit if someone would explore this effect in some mathematical way, and applied in a practical way.

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Are you related to BJ and how is he holding up? Should I send him some more books or what? Is he buff yet? How's his chess game?