How to strike a ball without throw in pocket billiards

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In pocket billiards, a throw refers to a phenomenon in which an object ball travels in a direction forwardly shifted from the intended direction when it is collided by a cue ball. Checking the magnitude of the shift is easy. Align two balls, in contact with each other, on a diagonal line from the front right side aiming at the far left corner pocket. Strike the nearer ball with a different ball, and the ball on the pocket side will be pocketed without fail.

However, if the different ball collides at the center of the nearer ball in a direction parallel to the longer cushion, the destination of the ball on the pocket side shifts to the right, away from the left corner pocket by the distance of about two balls, even though the collision point is the same. This implies that in most cases, a ball will not be pocketed even if it collides at a geometrically correct contact point. Hence, experts must be conducting some kind of countermeasures or corrective actions when they precisely pocket balls one after another.

Needless to say, the cause of a throw is friction at the time of collision. If there were no friction on the surfaces of balls, the only force acting on the collision point will be the force directed toward the center of the ball, which does not cause a throw. The force of friction on the surface combines with the force directed toward the center to shift the direction of the object ball.

The friction coefficient on the surface of a ball varies depending on the presence of materials adhered to its surface. Adhesion of moisture and oil is likely to occur in a billiard room, and it depends on the degree of air conditioning or the frequency of exposure to human hands. Therefore, the value of a throw is not alway the same.
This indicates that an orthodox method to calculate the value of a throw by analytically determining the velocity of a cue ball at the collision point against a collision point of a still object ball in the light of friction may end up in vain.

P-1 　Overlap and angle of the rotation axis that reset the horizontal velocity of a cue ball at a collision point to zero in a rolling collision

Fig P-1-1

　　　　　

Suppose that when a rolling cue ball collides with an object ball at an overlap of α with its tilt angle of the rotation axis θ, the horizontal velocity component at the collision point becomes to zero. The solution for the calculation is shown below.
Note that the following describes a case in which the right side of an object ball is aimed. Apply reversal for the left side.
When
　　　　　α　：　　The angle of the traveling direction of the object ball　
　　　　　θ　：　　The angle of the rotation axis of the cue ball at collision (equals to the angle of the strike point)　
then
　　　　　θ=arctan(sinα)

To correct the following figure, this formula was recalculated in early July 2015 for verification. The formula was found to be correct, with the same numerical relationship between θ and α given from a different method involving a graphical illustration.
The relationship is plotted on the graph below. The graph shows that α = 90° when θ = 45°, and inbetween figures are slightly larger than the proportional values.

Fig P-1-2
　　
　　　　　　　

When α = 90°, the cue ball touches the object ball and passes by as shown in Fig P-1-3 below. In this case, θ = 45° is obtained from the formula above. As shown in the figure below, the contact points of the cue ball and object ball are where the circumferential velocity becomes the same. Here, the ground velocity is zero and horizontal force is not produced on both balls.

Fig P-1-3

　　　　　　　

On the other hand, α＝0 is when the cue ball collides head-on with the object ball. The collision point of the cue ball is its forefront point. The horizontal velocity component of this point is zero if the rotation axis is completely horizontal. There is a large velocity component acting in the vertical direction, but it is not a force that shifts the object ball to the side.
Thus, when colliding against the right side of the object ball with the cue ball, the angle of the rotation axis that resets the horizontal velocity component of the cue ball at the collision point to zero, i.e. that does not cause a throw, is in the range of 0 =< θ < 45°.

The solution above is for the moment when a rolling cue ball collides against an object ball. Based on this solution, the condition of the moment of striking the cue ball with a cue should be determined. The condition for the case in which a slipping cue ball collides against an object ball is mentioned later.

P -2　The axis angle for the moment when a ball starts to roll after slipping, following a strike by a cue

A cue ball always slips right after it is struck by a cue, with one exception. The angle of the rotation axis changes as the ball slips, and after it stops slipping and starts rolling, the angle will be kept constant.

Once again, the relationship between the initial angle of the axis and the strike point angle are as shown in the figure below.

Fig P-2-1

　　　　　　　　

The figure shows that the angle of the rotation axis and the strike point angle are the same value (θ) at the moment of strike. The angle of the rotation axis takes an angle rotated clockwise from the horizontal line, and the strike point angle takes an angle rotated clockwise from the vertical line. Hence, when the strike point angle is ±180°, the strike point is right under the ball.

The off-center rate Sp refers to the distance between the strike point and the center of the ball. Sp = 1 is when the strike point is located at a point off-center by a distance of 40% of the radius, and Sp = 0 is when the strike point is at the center. When you strike a ball directly above the center at a point where Sp = 1, the ball does not slip and rolls from the beginning with the most stable movement of the direction.

Once again, the following figure, already shown in Fig 1-1 in other section, shows the changes in the rotation axis while the ball slips and then starts rotation, after being struck by a cue. In this figure, multiple results for a constant strike point and varied velocities are shown.

Fig P-2-2
　

Right after the cue ball is struck, it starts slipping, decreasing the axis angle, and once it starts rolling, it maintains the same axis angle. If the strike point is the same, the axis angle of a rolling ball will also be the same value regardless of the initial velocity.

　　　　　θ＝arctan((7・Sp・sin(θst)/(5+2・Sp・cos(θst))

　　　　　　　　　Sp　：　Off-center rate at shot
　　　　　　　　　θst：　Axis angle at shot (strike point angle)
　　　　　　　　　θ　：　Axis angle at the start of rolling

In this analytic solution, neither the initial velocity of the cue ball nor the slipping friction coefficient of the cloth of the table is included. Axis angle while rolling is determined by only the off-center rate Sp and the strike point angle θst, regardless of the strength of the strike and the condition of the table. Therefore, this axis angle becomes the angle at which the cue ball collides against an object ball.

θ and θst in the relational formula are plotted on a graph by numerical calculation in other section. It is partly revised and shown below. Here, Sp = 1.

Fig P-2-3
　
　　　　　
　As mentioned earlier, the axis angle of a rolling ball that resets a throw to zero is 45° or below. This figure shows that the corresponding strike point angle (axis angle) at shot is 59.34° or below. In this range, the graph shows values of an upper semi-circle. Roughly speaking, to collide a rolling cue ball against an object ball, the cue ball should be struck with a strike point angle of 60° or below, and an angle reduced by about 30% will be the axis angle of the cue ball at the start of rolling, i.e. the axis angle at the time of collision.

Meanwhile, there are some people who strike the lower part of a cue ball in pocket billiards. From the graph, it can be seen that there is a range where the axis angle of the rolling ball is 45° or below even when the strike point angle is around 160°. However, as you can see, the curve is steep and the result is unstable. You can say that striking the lower part of a ball to stabilize the direction of the shot may cause an unstable throw due to uncertain axis angle at the time of striking the object ball.

P -3　When a slipping cue ball collides against an object ball

Strongly striking a cue ball lengthens the slipping distance, and the cue ball is likely to collide against an object ball in the state of slipping. Controlling the axis angles at collision in the state of slipping is very difficult. This is mainly due to the unstable initial velocity.

Colliding in the state of slipping is the same as colliding before the ball starts rolling, and the axis angle at collision is in the middle of the strike angle and the axis angle of a rolling ball. This indicates that when striking an object ball with a slipping cue ball, the direction of the object ball does not significantly deviate from the intended direction if the object ball is collided by a cue ball struck with a strike angle smaller than the strike angle in which the cue ball is to be struck to collide against an object ball in the state of rolling, in consideration with the estimated distance that the cue ball slips.

P-4　Summary

This article gained analytical results as for a throw that has not been a subject to quantitative analysis so far in the field of pocket billiards, and suggested the conditions for actual games.

 Since this article reaches the conclusion via two theoretical formulas, a simple condition by purpose for striking a ball is not shown, but a practically effective strike point is understood.

 In sum, if the strike point is located away from the center of the cue ball by a length 40% of the radius, the cue ball should be struck at the upper right within the range of 60° or below when aiming at the right side of the object ball, and at the upper left within the range of -60° or greater when aiming at the left side of the object ball. The angle of the strike point should be determined inversely proportional to the overlap in that range.

Striking the cue ball at an angle outside these ranges should be avoided as much as possible. Note that the angle of the strike point should be slightly larger when the strike point is smaller than 40% of the radius (Sp < 1).