Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
New Phys.: Sae Mulli 2024; 74: 621-633
Published online June 28, 2024 https://doi.org/10.3938/NPSM.74.621
Copyright © New Physics: Sae Mulli.
Hyeong-Chan Kim^{*}
School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 380-702, Korea
Correspondence to:*hyeongchan@gmail.com
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the collision dynamics of a spinning cue ball approaching a static object ball with equal mass on a plane, common in billiards. While typical collisions in billiards are nearly perfectly elastic, with a restitution coefficient close to 1 and low friction, we explore three deviations from ideal elastic collisions: The non-elastic nature, the friction effects between the balls during collision, the friction between the ball and the table. We describe the detailed collision outcomes, emphasizing the importance of considering frictions. We reveal that friction, both between the balls and with the table, significantly influences the post-collision motions, deviating from the expectations of a purely elastic collision. The insights gained contribute to a better understanding of ball dynamics, impacting strategies and gameplay in billiards.
Keywords: Billiards, Collisions, Friction
Billiard players claim that when a cue ball having topspin collides head on with an object ball, the cue ball retreats briefly before moving forward due to friction with the floor. Individuals familiar with the perfectly elastic collision of two objects with equal mass find it difficult to intuitively accept this claim. However, upon examining super-slow-motion videos, such as those on Dr. Dave's YouTube channel[1], it is confirmed that, immediately after the collision, the cue ball briefly lifts off the ground, moves slightly backward, and then advances again due to rotation. In this way, collisions between two billiard balls exhibit phenomena that deviate from intuitive expectations based on perfectly elastic collisions.
To understand these phenomena, a comprehend understanding of collision dynamics is necessary. Theoretical studies on billiards have seen limited development since the early 1830s when Coriolis authored a book on the mathematical theory of spin friction and collision in the game of billiards, translated into English by Nadler[2]. Nearly a century later, Moore attempted the second physics analysis[3]. Subsequent research has covered various aspects, including collisions between billiard balls[4, 5], collisions with cushions[6], high-speed camera analyses[7, 8], and model studies for developing robotic systems for billiards[9, 10]. When using a cue to strike a billiard ball, the ball moves in a direction almost identical to the cue's striking direction. However, due to friction, the ball deviates slightly from the cue's direction, known as the squirt phenomenon[11-16]. Although the angle is small, this deviation is a crucial factor that makes it challenging to hit a distant ball with the desired thickness in actual gameplay.
The motion of a billiard ball is described by the translational motion of the center and the rotational motion around the center. When billiard players use the cue to strike the ball, the ultimate goal is to control the translational and rotational motion appropriately. Additionally, after the cue ball collides with the first object ball, the aim is to guide both balls' trajectories according to the desired outcome. Precisely placing the cue ball at the intended position on the object ball requires extensive training. However, in practical games like 3-cushion or 4-cushion billiards, using the cue to send the cue ball to the object ball (assuming not aiming for a perfectly accurate thickness) is relatively straightforward, excluding the squirt/curve phenomenon. Therefore, during the game, the most critical element is understanding how the moving ball (cue ball) will behave after colliding with the stationary ball (object ball). This study aims to address this aspect.
Firstly, two approximations that closely hold during the collision of two billiard balls are presented to assist in calculations. First, the friction between the balls during collision is considered negligible. The friction coefficient between billiard balls, denoted as μ, varies depending on the state and type of the balls but generally exists in the range of
Assuming no friction, angular momentum is not transferred between the colliding balls. Secondly, billiard balls undergo a perfectly elastic collision. The coefficient of restitution during the collision is approximately
A collision with a restitution coefficient of 1 is considered a perfectly elastic collision. Due to the coefficient of restitution being very close to 1, the collision between the balls closely resembles a perfectly elastic collision. For example, colliding a non-rotating cue ball head-on with a stationary object ball results in the cue ball to stop, and the object ball to move almost at the same speed as the cue ball before the collision. Furthermore, colliding a non-rotating cue ball with a non-zero impact parameter with a stationary object ball shows that the separation angle between them immediately after the collision is very close to
If gravity is disregarded, three factors are involved in the adjustment: friction between the colliding balls (proportional to the friction coefficient μ), deviation from a perfectly elastic collision (
The billiard table serves two roles during the collision of balls. Firstly, it provides normal forces to the balls during the collision that prevent them from falling. Secondly, it imparts frictional force to the balls, proportional to the vertical normal force. Frequently, it was assumed that significant changes in the vertical normal force do not occur during the collision to ignore the effect of this frictional force. If this assumption holds, the collision between two balls can be much simplified. On the contrary, if this assumption is incorrect, friction with the floor will affect the post-collision motion of the balls. In reality, billiard balls roll on a smooth cloth laid over a hard surface. Therefore, the object providing vertical support force to the balls is the hard surface, and the cloth plays the role of providing friction with the balls.
The collision time between two colliding billiard balls is approximately
For a collision with an initial speed of
These observations suggest that the assumption may not be appropriate that significant changes in the vertical normal force do not occur during the collision. On the other hand, if the frictional force lifts the cue ball, the reactional force tends to push the object ball downward. The magnitude of this frictional force is significantly greater than gravity, making it impossible to ignore. Then, the normal force exerted by the floor on the object ball cannot be neglected either. Additionally, the frictional force between the object ball and the table would also large, particularly due to the static friction present when the object ball is at rest before the collision. The static friction between the table and the ball, which brings the ball to a stop, is determined by the material of the table surface. Since the collision occurs in a very short time, the fiber material cannot respond during the collision. Therefore, the table and the ball experience the maximum static friction during the collision time. This static friction is significantly larger than other frictions making it a crucial factor that causes significant deviation from the results expected in a perfectly elastic collision.
The equations of motion for the collision between two balls with equal masses will be formulated in Section II. Section III will demonstrate how these equations are utilized during the collision process. Section IV will analyze the collision results, and finally, Section V will summarize the research findings in this paper.
Let the mass of the cue ball and the object ball be denoted as M, and their radii as R. Assume both balls move on a flat table, which is parallel to the (x,y) plane. Consider colliding the cue ball, which is in motion with a center velocity
In the following, we denote the center velocities of the cue ball and the object ball as
As illustrated in Fig. 2, the force
Thus, if there is topspin or draw on the cue ball (
Listing the forces acting on the cue ball,
where
Depending on the situation, if the vertical force becomes very large, the friction with the table cannot be ignored. However, it still acts as a second-order correction since its magnitude is proportional to the product of the two friction coefficients. Therefore, in this paper, we assume that the friction between the cue ball and the table can be ignored compared to
Similarly, for the object ball, the forces include the force exerted by the cue ball
The object ball stands still before the collision and then starts moving. As discussed in the introduction, the object ball, being supported by the soft fibers of the table, experiences motion only influenced by these fibers during the short moment of collision. In this brief moment, the displacement of the object ball is extremely small, to the extent that it is difficult to consider it moving through other fiber materials. Therefore, during the collision, the frictional force between the object ball and the table is given by static friction. The maximum static friction coefficient^{2} is approximately
This value, being relatively larger than the coefficients of kinetic friction or rolling friction, cannot be ignored, and its influence is notable in the context of the interaction with the table. The relative velocity,
where
where
As discussed in the introduction, if the cue ball is moving rapidly
The subsequent calculations will utilize this approximation.
Following the discussion above, the equations of motion for the changes in the center velocities of the cue ball and the objective ball can be written as:
where
where,
The frictional forces between the cue ball and the object ball act in the opposite direction of the relative velocity at the point of collision. To calculate the relative velocity at the point of collision for each ball, let's denote the displacement from the center of the cue ball to the collision point as
Conversely, the relative velocity of the cue ball with respect to the objective ball at the contact point is
Here, the initial directional angle
Because the friction acts along the opposite direction to the relative velocity, the angle satisfies
The time-dependent change in the relative velocity is then given by
where we use Eqs. (9) and (10).
Let's denote the normal (perpendicular to the collision plane) impulse exerted on the balls during the collision as
This value monotonically increases during the collision. We will use this impulse in place of time. Define the impulses in the directions perpendicular and parallel to the collision plane during the short time dt as
respectively. Now, the equations of motion (9) can be expressed by multiplying both sides by dt and using the defined expressions as follows:
Regarding the change in angular velocity in (10), when expressed in terms of impulse, it becomes:
Here, ≈ indicates the approximation neglecting gravity and friction between the cue-ball and the table.
The change in relative velocity with respective to the impulse can be obtained as follows:
Here, recalling that
and utilizing
As observed from this equation, the angle θ, which indicates the direction in which frictional force is acting, changes over time. Because
In both cases, there is no event during the collision that reverses the sign of the angle θ. Moreover, as evident from the equations of motion, the friction coefficient
In this section, we aim to provide a detailed chronological description of the process that occurs at the moment when billiard balls collide. To illustrate the sequence of physical phenomena during the collision process, we focus on the impulse
The moment the cue ball reaches the object ball, both balls enter a state of sliding against each other. At the moment of collision, the cue ball simultaneously pushes the object ball vertically with a speed of
Here, we assume that the friction coefficient is independent of the sliding speed or vertical pressure, which is valid as long as the sliding speed on most surfaces is not too fast. In collisions between balls, the friction coefficient μ is not large, making it difficult to enter a stick state where sliding stops. During sliding, changes in the horizontal and vertical components of relative velocity are given by, from Eq. (17),
As mentioned earlier,
Due to the step function, the equation (23) needs to be solved separately for each case where θ is negative or positive. Fortunately, the results for the case of
Thus, the magnitude of the component parallel to the collision plane of the relative velocity is given by
and
From the initial values (11), it follows that
Using this, the first two equations in Eq. (23) become
where
The differential equations (27) can be integrated using hypergeometric functions:
Here,
and
Here,
The results for positive θ can be obtained by taking the limit as
As the relative vertical velocity decreases, the two balls are compressed, and energy is accumulated. Let's denote the vertical impulse at the moment when this vertical compression stops, i.e., when
By solving the first equation and the second equation for k=z, we can determine the value of
The total energy absorbed during the compression process can be obtained by integrating the force acting in the vertical direction over the displacement.
If the vertical component of the relative velocity varies linearly with the impulse during compression, the magnitude of the work done as the vertical impulse increases from 0 to
In the case of
At the moment when the vertical velocities of the cue ball and the object ball become equal, both balls stop compressing and enter the restoration process. During this period, as the shapes of the two balls return to their original state, the elastic energy accumulated during the compression process is utilized to push each other away. Let's denote the final value of the vertical impulse as
The final horizontal relative velocity is given by (29):
Therefore, the current assumption that the friction is small enough not to reach a stick state still holds if
where, in the second equality, the first equation of (32) is used. If θ is positive,
The elastic energy restored during the restoration process can be calculated using the above equation:
From the above results, the ratio of absorbed elastic energy during the compression process to the restored energy can be related to the square of the (energetic) coefficient of restitution, given by:
Here,
From this equation, one can write the value of
Generally, although this equation presents a complex relationship, in the case of
In general,
Using the above results to find the post-collision velocity of the cue ball, with the help of (17) and (A1), we have:
where
In the y-direction, the law of conservation of momentum holds, but it does not hold in the x and z-directions.
This phenomenon occurs due to the counteracting effect of vertical forces, neutralizing the impact of frictional forces.
The angular velocities of the two balls after the collision become
as derived from (18). Utilizing the above results to compare the energy before and after the collision, one may notice that the conservation of kinetic energy does not hold when there is friction and the coefficient of restitution is not 1.
In the previous section, we discussed the collision between two balls in a general form.
However, in the case of a collision between billiard balls, the magnitude of the coefficient of friction μ between the balls is not large, resulting in a small change in the relative velocity in the direction parallel to the collision plane.
In such cases, it is sufficient to use linear approximation by expanding the function
Firstly, from the definition of the function
We have described up to quadratic approximation, as it is necessary for expanding the coefficient of restitution formula (37) to the linear order in the friction coefficient μ. Conversely, we can also use
First, we calculate
Using Eqs. (29) and (43), we get
The calculation process uses Eq. (20) and Eq. (26) also.
If
In future calculations,
Then, let's solve for
Therefore, the following holds:
Now, using the above results and the coefficient of restitution formula (37), let's determine
Here,
The elastic energy recovered in the restoration process can be obtained from (36). Using a similar process as above, we get:
Now, using (37), we have:
Here,
In other words, up to the linear-order approximation in μ, the vertical impulse is determined by the coefficient of restitution. Therefore, using (46) and (11), the final vertical impulse is given by:
Comparing this equation with (38), the impulse correction is:
Now, we write the velocities of the cue ball and the object ball after the collision given by Eqs. (39) and (40):
In the case where there is a draw, i.e.,
Now, let's examine the changes in angular velocity for the cue ball and the object ball. Firstly, there is no change in angular velocity component in the direction perpendicular to the collision plane. On the other hand, the angular velocity components along y and z-directions change according to Eq. (41):
This means that the object ball obtains a larger angular velocity with a higher initial horizontal speed and a larger friction coefficient. Additionally, we observe that the friction between the object ball and the ground only contributes to the change in the angular velocity of the object ball when there is a draw.
In this work, we discuss the phenomenon that occurs when a cue ball, rotating on a plane, approaches and collides with a static object ball of equal mass. In the case of billiard balls, the coefficient of restitution is close to 1, and the friction coefficient takes a value close to 0, demonstrating a collision phenomenon close to a perfectly elastic collision. In this scenario, immediately after the collision, the trajectories of the cue ball and the object ball form a right angle with each other. The cue ball moves in the direction parallel to the collision plane, while the object ball moves in the direction perpendicular to the collision plane. This situation is illustrated in Fig. 3 using red arrows, where
In this paper, three effects which develop deviations from the results of a perfectly elastic collision in the collision of balls are discussed. These three effects are: 1) the collision between balls is not perfectly elastic, 2) the effect of friction between balls during collision, and 3) the effect of friction between the ball and the table at the moment of collision.
Before discussing the detailed results of all these effects, let's first consider the scenario where a cue ball with topspin collides head-on with an object ball, taking into account the presence of frictions, the topic we mentioned at the beginning of the introduction. Firstly, assume that the cue ball has only rolling motion before the collision. Therefore,
Now, let's describe the detailed results of the collision. Firstly, the phenomena arising from the difference with a perfectly elastic collision, due to the absence of friction, are proportional to
Ignoring the motion perpendicular to the billiard table and comparing the results with a perfectly elastic collision, the differences in the speed and the direction of the object ball are proportional to
If we interpret
As seen in Eq. (39), without friction, the object ball moves along the x-direction, perpendicular to the collision plane. If there is friction, the object ball's velocity has components in both the y-direction and the vertical upward z-direction parallel to the collision plane. The magnitude of the upward velocity component is given by:
This vertical motion is linearly proportional to the friction coefficient and occurs only when the cue ball has an angular velocity about the y-axis, i.e., there is topspin. The directional difference is denoted by ψ in Fig. 3 is approximately given by:
This angle is maximized when there is no initial rotational angular velocity about the y-axis, and for a given friction coefficient according to Eq. (1), it varies by approximately
When two balls of equal mass undergo a frictionless perfectly elastic collision, the cue ball moves in the y-direction after the collision. Neglecting the vertical upward motion, the motion of the cue ball, as seen from the first equation in (52), shows two changes. First, due to the effect of friction, the speed in the y-direction parallel to the collision plane decreases. Second, as the collision is not perfectly elastic, the cue ball gains a velocity component in the x-direction perpendicular to the collision plane. As a result, the cue ball deviates from the y-direction by an angle
If the cue ball engages in a perfectly elastic collision (
As time progresses after the collision, the angle formed by the velocities of the two balls changes due to the rotational motions. The post-collision trajectories of the balls are determined by the velocities/angular velocities resulting from the collision and the friction with the table. If there is angular velocity around the axis of the ball’s heading direction, the ball follows a curved path. In the case of the object ball, the rotation induced by friction is not significant, and bending is not visibly apparent. However, the cue ball, depending on the situation, can exhibit significant curvilinear motion. Further research is needed to investigate these trajectories in detail.
The results of this study contribute to the understanding of ball movements and collisions in billiards, providing insights into strategies and gameplay. Additionally, this research can offer valuable information in the field of physics education and experiments.
We attach an integration formula which is used frequently in the text:
In deriving this formula, we us Eq. (23).
This work was supported by the National Research Foundation of Korea grants funded by the Korea government RS-2023-00208047 and Korea National University of Transportation Industry-Academy Cooperation Foundation in 2023.