Ball Motion Properties in Stun and Follow Shots
It happens in life, as in grammar, that the exceptions outnumber the rules
Remy de Gourmont
Anyone who has had the opportunity to learn the basics of billiards – whether Pool, Snooker, Carom, or Russian Pyramid – certainly knows the two basic rules which describe the motion of balls after collision. These are the so-called 90- and 30-degree rules.
This study attempts to demonstrate how these rules work for Russian Pyramid using high-speed video.
The principle of the 30-degree rule can be clearly seen in the training videos made by David G. Alciatore («Dr. Dave»): http://www.youtube.com/watch?v=wZIX4WLd_JQ.
It should be noted that this rule is valid only for a cue ball which rolls on the table without slipping. The essence of the rule is that the cue ball deflection angle (the angle between the initial and the final directions of motion) does not depend on the speed of the cue ball, its size and weight, or the value of friction between the ball and the cloth, and is solely determined by the cut distance, that is, the distance between the centers of the cue ball and the object ball at the moment of contact, when viewed along the shot line).
According to the studies of Dr. Dave, for a wide range of ball-hit fraction (1/3–3/4), the cue ball deflection angle is close to 30 degrees. The maximum value of cue ball deflection angle (33.7 degrees) is achieved when the cut shot is made at a little more than a half-ball distance (0.53), that is, when the distance between the centers of the object ball and the cue ball (when viewed along the shot line) at the moment of their contact equals half the diameter of the ball.
The 30-degree rule has special significance in the case of Russian Pyramid due to the legality of the cue ball potting. Moreover, in some types of modern Russian Pyramid, a potting of the cue ball is the main tactical component of the game, while half-ball shots with varying impact force constitute a basic technique of the game.
Not so long ago a “simulator of the positions of golden zones”, designed by Anatoly Kalyuzhny, appeared on the website of the Billiard Amateurs League (http://www.llb.su/node/951852). This simulator is a template which, according to its author, “makes it possible to establish benchmark positions for striking the cue ball through the golden zones with high accuracy”. When mr. Kalyuzhny was creating the simulator, he used his own calculations, describing the motion of the cue ball after collision with the object ball. The results of these calculations for one particular case of the follow shot were presented in the form of a graph which, according to the author, is “correct” as opposed to a similar graph shown in Dr. Dave’s works (http://lastball.narod.ru/tp33/tp33_05.jpg).
It is worth noting that Anatoly Kalyuzhny applied a non-conventional designation of the cut shot magnitude. Typically, the ball-hit fraction for straight-in collisions is considered to be 1, while the maximum thin cut is considered to be 0.
For verifying the declared “correctness”, cue ball deflection angles depending on the cut of the object ball, was measured by using a high-speed camera. The strokes on the ball were made using a special unit (see http://dbkcues.ru/articles/cuetestingunit-2/). The force of the stroke corresponded to a cue ball velocity of 2.5 m/s. The changes in the magnitude of the cut were done by moving the object ball in conductor, while the cue ball was set in one and the same position using a prism.
The high-speed camera was attached to a girder crane and was positioned strictly vertically over the billiard table. The shutter speed was 3200 fps. The frames of high-speed footage were used for determining the final angle of cue ball trajectory for each of the 68 kinds of collisions with the object ball. Measurement results are presented as a graph which also shows the results of the calculations made by Anatoly Kalyuzhny (blue dashed line) and Dr. Dave (red line).
As can be seen, the experimental data with certain accuracy reproduce the results of Dr. Dave. The trajectories of cue ball and object ball were produced on the basis of high-speed video for the ball-hit fraction of 3/4 (where the greatest difference between the results of Dr. Dave and A. Kalyuzhny were observed): http://www.youtube.com/watch?v=JZlGw3l01UU.
The measured angle in the trajectory of the cue ball for this magnitude of cut was 29.1 degrees. In the calculations of Dr. Dave, this angle was about 28 degrees, and Anatoly Kalyuzhny claimed the angle to be over 34 degrees, which is not true.
Thus, the equations used by Dr. Dave quite accurately describe the actual motion of the cue ball, despite the fact that they were obtained under the assumption of perfectly elastic collisions of balls and the absence of friction between them. To be fair we should say that more exact calculations using the coefficients of friction and restitution do not lead to significant changes in the deflection angle of the cue ball. Thus, in the work, “The Effects of Ball Inelasticity and Friction on the 30° Rule” (http://billiards.colostate.edu/technical_proofs/new/TP_A-6.pdf) Dr. Dave has shown that for the half-ball cut, considering the coefficients of friction and restitution leads to an increase of only 0.3 degrees for that angle.
The principle of the 90-degree rule as well can be found in the training video of Dr. Dave (http://www.youtube.com/watch?v=DU5UkMFPDrA).
The essence of this rule is that the cue ball and object ball depart at an angle of 90 degrees in the case of the stun shot (that is, when the cue ball prior to the collision with the object ball is sliding on the cloth without any rotation) regardless of the magnitude of cut. In this case the object ball would move along the line connecting the centers of the balls at the moment of contact, while the cue ball continues its motion along the tangent line.
Unlike the previous rule, it is not difficult to establish the 90-degree rule; it is enough to decompose the momentum of the cue ball at the moment of collision into normal and transversal components or use the laws of the conservation of momentum and energy in the same way as it was done in the work of Dr. Dave (http://lastball.narod.ru/tp31/tp31.html).
It should be noted that the determination of the 90-degree rule is based on two important assumptions: there’s no friction between the balls, and the collision is perfectly elastic. As we can see from the following video, in fact, there is friction between the balls.
The same applies to the assumption concerning the perfectly elastic collision of balls.
As will be shown below, both friction and the absence of perfect elasticity during the collision of balls, under certain conditions result in substantial violation of the 90-degree rule.
INFLUENCE OF FRICTION BETWEEN THE BALLS ON THE MOTION OF CUE BALL
Many players must have faced a situation when the attempt to cut a simple ball near a pocket mouth at a soft stroke results in not cutting it “all the way,” with the ball bouncing off the cushion tip and becoming exposed for the stroke of the opponent. Besides the lack of skill, this annoying slip has a rational explanation. In the literature this phenomenon is called “the throw”. This effect occurs when friction force operates at the moment of collision between the cue ball and the object ball, which leads to the situation when the object ball after collision with the cue ball does not move along the line connecting the centers of the balls, but diverges at a slightly narrower angle, that is, gets “undercut”. The influence of friction force on the motion of the object ball is manifested not only during stun shots, but also during strokes with cue ball rotation. Moreover, the sidespin of the cue ball depending on the direction can both increase the angle of throw and compensate for it. All these effects are described in great details in the article of Leonid Baltsev (http://billiard.net.ua/magazine/index.php?id=972).
However, that article contains one significant inaccuracy in the following assumption, “According to the laws of physics, friction force is practically independent of the relative velocity of the rubbing surfaces. This means that the angle of the throw does not depend on velocity.”
Indeed, the course of general physics gives the idea that dry sliding friction does not depend on slipping velocity of rubbing bodies. However, in fact, there are so many materials and processing methods that the friction coefficient can both increase and decrease depending on velocity. Experimental data for brake pads of trains can be a good example of the reduced friction coefficient (http://www.pomogala.ru/tormoza/tormoza_2.html).
The friction coefficient of billiard balls behaves in a similar way. In his article (http://billiards.colostate.edu/technical_proofs/new/TP_A-14.pdf) Dr. Dave referred to the experimental results of Wayland Marlow, who measured the dependence of friction on velocity.
Marlow determined the behavior of the friction coefficient using just three experimental points, which in general is not a very correct method. Similarly defined friction coefficient for Belgian balls Aramith which are used in Russian Pyramid, has turned out to be slightly larger at all implemented velocities. This may possibly be explained by the use of another material in the manufacturing process of balls, or by the wear on the surfaces of test balls. Nevertheless, whatever the absolute values of friction coefficient were, their obvious dependence on velocity would necessarily lead to the change in the deflection angle of the object ball. And it is not the forward speed of the cue ball that matters, but the relative slipping velocity between the contact points of the balls (this velocity is the sum of the tangential component in the forward velocity of the cue ball and the linear velocity of its rotation). In other words, the throw angle of the cue ball will depend on impact force, cut magnitude (cut angle), and on the presence of rotation (including follow and draw).
A trajectory of the object ball at a stun shot is presented below on the basis of high-speed video. The forward speed of the cue ball is 2.56 m/s; the cut angle is 23.5 degrees.
It is easy to notice that due to the friction force, the object ball becomes “undercut” at the angle of 4 degrees. At this throw angle, at a distance of one meter the object ball would deviate from the line connecting the centers of the balls at the moment of their contact to a distance greater than the diameter of the ball.
With a similar stroke using a follow shot when the speed of the cue ball was 2.61 m/s, the angle of undercut for the object ball became an order of magnitude less.
This is related to the increase in the slipping velocity of the contacting surfaces of balls (and as a consequence, friction reduction) due to the presence of angular velocity.
If we trace the dependence of the cue ball throw angle on the cut angle, we will have the following picture for stun shots and follow shots:
The red dotted line corresponds to the calculation of stun shots, made according to the equations of Dr. Dave (http://billiards.colostate.edu/bd_articles/2006/sept06.pdf) and friction coefficients taken from Marlow’s experiment. The top black dashed line is the same calculation, but the friction coefficients were taken for the balls used in Russian Pyramid. Black circles are the experimentally measured throw angles of the object ball during stun shots. The lower black dotted line and the triangles underneath correspond to the calculation and experiment during follow shots.
If we now look at the measured data of object ball deflection angles during stun shots, we can see that the graph has distinctive kinking with a pronounced maximum of the object ball deflection angle, which amounts to about 4 degrees when the speed of the cue ball is 2.5 m/s. Dr. Dave and Bob Jewett have previously obtained similar data: http://i.pixs.ru/storage/4/9/1/ThrowDeyvp_7444256_9228491.jpg http://www.sfbilliards.com/throw.gif
The kinking on the graph is the result of the changing nature of friction between the balls.
If the cut angle is sufficiently large (thin cut), the slipping of cue ball relative to object ball occurs at the point of contact for the entire time of collision. This allows us to determine the throw tangent for the object ball as the ratio of the tangential and normal components of the force acting on the object ball, which is ultimately equal to the coefficient of friction Ft / Fn = k ∙ Fn / Fn = k. Since the coefficient of friction depends on relative velocity of the balls at the point of their contact, which in turn depends on the cut angle, the throw angle of the object ball will also depend on the cut angle, which may be observed on the graph at cut angles greater than 25 degrees.
If the cut angle is small (thick cut), friction force during the collision has time to equalize the instantaneous velocities of cue ball and object ball at the point of their contact (Vcb – R ∙ ωcb = Vob + R ∙ ωob). This condition eventually makes it possible to determine the tangent of the throw angle for the object ball, which will amount to 1/7 ∙ tgϕ, where ϕ is the cut angle.
Thus, the throw effect caused by the friction between the balls under certain conditions may significantly reduce the angle of deflection of the object ball (at least for 4 degrees). This would lead to the fact that the cue ball and the object ball would depart at an angle of less than 90 degrees. Weakening of the throw effect with a sufficiently large cut can be achieved by increasing the force of stroke or rotating the cue ball around the horizontal axis.
INFLUENCE OF THE COEFFICIENT OF RESTITUTION ON THE MOTION OF THE CUE BALL
The throw effect of the object ball, as indicated above, is based on the operation of friction force. During the interaction between the balls, friction force creates a small momentum increment dP = Fтр ∙ dT, which determines the deviation of the object ball. It is obvious, that exactly the same friction force acts from the object ball upon the cue ball, but unlike the the object ball, momentum increment for the cue ball coincides with the direction of its deflection. This means that friction force for the cue ball may only change the velocity of deflection, but not its direction / trajectory.
However, if we go back to the picture showing cue ball throw effect for the stun shot, and try to determine the angle between the direction of cue ball deflection and the line connecting the centers of balls at the moment of their contact, we may see that the angle is not 90 degrees, but only 84.8 degrees. In other words, there was a decrease in the angle of cue ball deflection by 5.2 degrees.
As it turned out, the decrease in the angle of cue ball deflection is even more pronounced at thicker cuts.
It is worth noting that after collision with the object ball in the thickest cuts, the cue ball passes a distance of about one centimeter, which causes some problems with accuracy of determining the angle of cue ball deflection.
We do not necessarily have to use a high-speed camera for seeing a decrease in the angle of cue ball deflection. It is enough to conduct a simple experiment which would require three balls and a table with markings. Two “frozen” balls are set on the line parallel to the short rail, while one of the balls at the same time is set on the central line parallel to the long rail. Using a stop shot we strike the cue ball at the ball set at the intersection of two lines, and observe its movement relative to the center line. The thicker the cut at the frozen balls is, the stronger an effect can be observed.
Since this phenomenon has not previously been described in Russian billiard literature, for my own use I called this effect “the narrowing of cue ball deflection”. At the same time, international literature, for example, Bob Jewett’s article “Pythagoras and Pool Perpendiculars” (Billiards Digest February, 1995 http://www.sfbilliards.com/articles/1995.pdf), does not only refer to this phenomenon, but contains its fairly detailed explanation. The comment by the author concerning the balls made of ivory is interesting, which manifest the effect of the narrowing in a more pronounced manner than in the case of modern balls.
The cause of this phenomenon is that the collision of the cue ball and object ball is not perfectly elastic. As we know, the property of elasticity is described by the coefficient of restitution (for an perfectly elastic collision the coefficient of restitution is 1, and for an perfectly inelastic collision it is 0). If the the coefficient of restitution is less than one, there is a situation when the cue ball after the collision with the object ball retains a part of its motion along the line connecting the centers of the balls. This residual motion causes the deviation of cue ball trajectory from the direction of the tangent, leading precisely to the narrowing of cue ball deflection angle.
More information about the narrow effect can be found in the work of Dr. Dave (http://billiards.colostate.edu/technical_proofs/new/TP_A-5.pdf), which contains equations for calculating deflection angles both for the cue ball and object ball using the coefficients of friction and restitution (NB!: a coefficient of 1/7 is missing in equation 10 before the sine, while in the work http://billiards.colostate.edu/technical_proofs/new/TP_A-14.pdf, a similar equation 14 does contain this coefficient).
The comparison of the experimentally measured narrowing of cue ball deflection angles for different values of cut with numerical calculation (red dashed line) using the corrected equations Dr. Dave can be seen below.
Here, the angle of the narrowing of cue ball deflection was counted between the tangent line drawn at the point of balls’ collision and the real direction of the ball deflection. The velocity of the cue ball at impact was 2.5 m/s. For the calculation we used the values of the coefficient of restitution, found in the analysis of a high-speed video for different cut angles. As can be seen from the graph below, the coefficient of restitution remained almost the same (0.92) for the entire range of cut angles, and it showed a slight decrease only at a very thin cut. Apparently, this decrease is related to the loss of energy, caused by the mutual unwinding of the cue ball and the object ball.
Cue ball deflection narrowing results in the fact, that the deflection angle itself has nonlinear dependence on the cut angle. As is seen from the following graph, the ball deflection angle at the stun shot has a clear maximum of 69 degrees reached at the ball-hit fraction of 5/6.
In conclusion, we may say that the 90-degree rule in fact does not work for any angle of the cut shot. As is shown above, this violation is a visible manifestation of two effects: the effect of the throw of the object ball and the effect of the narrowing of the cue ball deflection. The first of the effects emerges from the friction between the object ball and the cue ball, while the second by the imperfect elasticity of balls.
The violation of the 90-degree rule is particularly strongly manifested at small angles of cut, where the value of the spreading angle of the object ball and the cue ball falls to zero.