Oblique Collisions Between Billiard Balls for Introductory Physics

The application of physics to billiards has a long history. For example, Coriolis wrote a book on the topic in 1835.¹ An oblique (not head-on) collision between the cue ball and another ball is an interesting example for an intermediate-level course on classical mechanics. At first glance, such collisions seem simple, but there are some subtleties. I will explore what can be reasonably covered in an introductory course without too much mathematics, emphasizing cases where the results are relatively simple.

Equation (4) implies that the triangle formed by the velocity vectors is a right triangle, because their lengths satisfy the Pythagorean theorem. In other words, the angle between the final velocities is 90°.

Does this simple result apply to billiard ball collisions? Wallace and Schroeder found that many textbooks used to claim that for an oblique collision between a cue ball and a stationary object ball, “the resulting angle between the directions of travel of the balls after the collision is 90 deg.,”² but they noted that this is “true only under rather exceptional circumstances on a pool table, as many pool players can attest.” It is a reasonably good approximation that the collision between the hard spheres is elastic. Also, any external forces can be approximately ignored during the very short collision. Therefore, immediately after the balls collide, the angle between their velocities will be close to 90°. However, the direction of the cue ball usually changes because of its spin and friction between the balls and the table. The exception is when the cue ball is sliding without rotating as it collides. A detailed analysis of the path of the cue ball after a collision with an object ball is beyond the scope of most introductory courses, but the behavior can be explained qualitatively.

Ignoring for the moment the frictional force between the balls, the two previous results can be combined to explain the paths of the cue ball after an oblique collision. Figure 3 shows the theoretical paths of the balls found by Wallace and Schroeder if the cue ball is rolling without slipping when it collides with the object ball.² (Onada found some simpler approximations for determining the final direction of the cue ball.⁵) Immediately after the collision, the balls move at 90° from each other, and they are slipping along the table. The cue ball will still have the same rotation about the x-axis just after the collision. Friction will cause the object ball to start rolling, but will not change its direction. The frictional force on the cue ball points somewhat opposite to the direction of motion because the cue ball is sliding and somewhat in the y direction because of its rotation. The net frictional force will cause the cue ball’s path to curve in the direction that it was initially moving, so this is considered a follow shot. Eventually, the cue ball rolls without slipping, so its path becomes straight. The sum of the final deflection angles (θ_o and θ_c in the diagram) will be less than 90°. If the cue ball has a backspin as it collides, its path will curve the opposite way, which is called a draw shot.

The coefficient of friction between the billiard balls is less than that between a ball and the table, but it can still have a noticeable effect. As shown in Fig. 4, there will be a frictional force f_o on the object ball in the direction that the cue ball initially moves after the collision. Due to this force, the object ball will not travel exactly along the line connecting the centers of the balls, but at a slightly smaller angle from the original direction of the cue ball as shown in Fig. 4. This effect is known as collision-induced throw, but it is sometimes just referred to as throw. The size of the throw angle depends on the direction of the object ball. The farther to the side the object ball is going, the larger the throw angle will be. Whether or not the throw angle must be accounted for depends on the distance to the intended pocket. If the object ball doesn’t have far to travel, the overall deflection won’t be large. The effect of the cue ball having spin about a vertical axis, or English, and the transfer of spin are interesting advanced topics, but are beyond the scope of the article.

There are collisions between two billiard balls where the simpler result of 90° between the final velocities is a better approximation. This occurs when two object balls are touching, which are referred to as frozen balls. In this case, shots can be treated as two separate collisions: the cue ball with the first object ball and the first object ball with the second one. When the cue ball strikes the first object ball, it typically doesn’t transfer very much spin because the surfaces of the balls are smooth. Since the first object ball strikes the second one with essentially no spin, the two object balls will move off at very nearly 90° from each other. The second object ball will move approximately along the line initially passing through the centers of the two, and the first one will move approximately perpendicular to that.

Players should be aware of two shots with frozen balls that are easy to sink. The first is the frozen ball combination⁶ or dead-in shot⁷ where the line between the two object balls goes toward a pocket. As shown in Fig. 5, the second object ball will go approximately along the connecting line, but will be thrown slightly to the opposite side of the line from where the cue ball strikes the first object ball. Second is the kiss⁶ or carom⁷ shot, which is shown in Fig. 6. In this case, a line perpendicular to the connecting line and passing through the first object ball is directed toward a pocket. If the frozen balls are lined up correctly, this shot is almost impossible to miss if the cue ball is shot from the quadrant shown, unless significant spin is transferred from the cue ball.

An interesting exercise is to look in billiards books for trick shots that involve frozen balls. Two trick shots from Mizerak and Panozzo⁶ are good examples. For the “Butterfly,” shown in Fig. 7, six object balls are carefully positioned so that they can be easily sunk in a single shot. The cue ball strikes balls 1 and 2, which collide with balls 5 and 6, respectively, sending them toward the pockets on the left. Balls 1 and 2 carom into balls 3 and 4, respectively, sending them toward the pockets on the right. Balls 1 and 2 then carom at right angles to the lines between them and balls 3 and 4, respectively, which sinks them in the side pockets. This shot doesn’t require any special skill if the object balls are set up correctly. A special rack is available for setting up the butterfly shot and a few other trick shots.⁸ A second example is “Easy As 1-2-3,” which is shown in Fig. 8. If the balls are placed carefully, it is easy to sink the frozen balls. Ball 1 caroms off the upper bumper to the lower side pocket, while ball 2 travels directly to the corner pocket. The difficult part is that the cue ball must be shot accurately and given left English so that it banks off the three rails to sink ball 3. Steve Mizerak used a more complicated variation of this shot in which six balls were sunk at the end of the Lite Beer commercial that made him famous.

In order to test their understanding, students can be given the following question based on an example by Koehler.⁷ Suppose that three object balls are frozen together as shown in Fig. 9. How should the cue ball be shot to sink ball 1? Assume that the cue ball will be rolling without slipping when it strikes an object ball. The answer is given in the endnotes.⁹ 

Wallace and Schroeder² concluded that, “billiard ball collisions on high-friction surfaces do not provide a particularly apt illustration of elastic collisions for introductory students.” I hope that readers will be convinced that these collisions can be discussed in introductory classes, especially collisions between frozen balls.