The Myth of the Tangent Line Part I

The Myth of the Tangent LinePart IRemember when you learned in high school physics class that all objects, regardless of mass, fall the same distance in equal times under the influence of gravity? Something didn't seem quite right about this, so you went home, stood on a ladder, and dropped a penny and a feather at the same time from equal heights. They don't fall at the same speed. Wondering why your physics teacher would lie to you about this, you confronted him with your experimental results the next day, only for him to remind you of the condition called “In a Vacuum.” All objects fall at the same speed in a vacuum. Physics is notorious for imposing near-impossible conditions for the sake of teaching physical principles at an elementary level and providing a reference point. Textbook problems always seemed to require point masses (masses that occupied no volume), perfectly elastic springs, inextensible strings, zero friction, perfect vacuums, and balls that slide but don't roll. As a graduate assistant, I worked in a college physics lab for two years and never did find the drawer of point masses and frictionless blocks. As a result, we start our analysis of a physical event with what theoretically happens under ideal conditions and then adjust for the reality of a particular situation. I'll tell you what happens in a vacuum, and then you will be able to calculate what happens in all other conditions (at high elevation, under water, on the moon, etc.).
The tangent line rule in pool is a similar ideal condition reference point. It is intended to be the reference point of the path of the cue ball after it collides with an object ball. Rarely is it the actual path until a high level of cue ball control is achieved. The tangent between two balls is a line perpendicular to the line joining the centers of the balls at the point of contact as shown in Diagram 1. The tangent line rule simply says that after collision, the cue ball and the object ball will depart at 90 degrees to each other. When all the conditions exist for this to be true, it is valuable information for the following several reasons:1. Depending on your ability to see a 90 degree angle on the table, it helps you determine the risk of a scratch. You know the path required for the object ball to its target pocket. An imaginary line drawn 90 degrees to that is the cue ball path. If a scratch is imminent, as in Diagram 2, you need to find a preventive adjustment. 2. In planning your next shot, the 90 degree rule gives you an indication of the direction the cue ball will take. Sometimes getting good position is just a matter of the right speed so the cue ball stops at the spot you want it along the 90 degree path, as in Diagram 3. 3. Sometimes the objective is to not only pocket a ball but also break up a cluster of balls with the cue ball in order to continue a run. With some good planning, you can position yourself on a shot so the 90 degree cue ball path is aimed at the cluster, as in Diagram 4. 4. Other times, especially in 9 ball, the 9 is right at the edge of a pocket, but a combination to bump it in is not possible. However, there may be a clear path for the cue ball to carom off the lowest numbered object ball (the 5 ball, for example) as in Diagram 5. To find the correct point of contact on the 5 ball, reverse the right angle process by drawing a mental line from the 9 ball back to the 5 to determine how much of the 5 ball to contact in order for the tangent line to point to the 9. The goal here is not to pocket the 5 but to carom the cue ball into the 9 for a win. Occasionally, you will find yourself in the fortunate position where you can pocket the 5 AND carom into the 9. This two-way shot is similar to Case #1 but has a double safety net in that the 9 helps prevent the scratch, and pocketing the 5 allows you to continue shooting even if you fail at the original goal (the 9). So where's the myth in all this, and if it is a myth why am I propagating it? Just as the physics teacher starts with the perfect vacuum condition for falling objects, pool instruction starts with the perfect condition for collisions, i.e., sliding cue balls. However, pool balls don't slide, at least not for very long. They roll. This ideal condition (sliding) provides a reference point from which you can predict cue ball paths for all the different kinds of spin you know how to put on the cue ball: draw, low left, follow, high right, massé, etc. Everything becomes relative to the sliding-ball tangent line. The root of the myth is that a very specific condition must exist for the 90 degree departure to take place. The condition is that the cue ball can have no angular momentum at the time of contact with the object ball. In other words, the cue ball must be sliding with absolutely no rotation. But round things prefer rolling to sliding. Rolling friction is much smaller than sliding friction so that rolling is literally the path of least resistance. Even when a sliding condition is achieved, it can be maintained for only a fraction of a second. A sliding cue ball very quickly becomes a rolling cue ball because of the friction between ball and cloth. To see this, place a striped ball on the foot spot with the stripe oriented perpendicular to the length of the table. Strike this striped ball as if you were striking a cue ball with draw, hitting it a cue tip below center and with moderate speed. You should observe the stripe spinning backwards at first then transitioning to a forward spin. The transition time is the only time the ball is sliding without rotation. This is similar to a piston changing directions at the peak of its stroke where, for a very short time, the piston is not moving. If the cue ball contacts an object ball during this transition, you get 90 degree departure. Even then, the cue ball follows the tangent line only until the cloth/ball friction causes the ball to start rolling again. Since rolling friction is much smaller than sliding friction, the cue ball very quickly finds its rolling “path of least resistance.” As soon as it starts rolling again, it leaves the tangent line. The same effect can be seen when a rolling cue ball collides with a stationary object ball. The reason for this will be examined in Part II of this series, but for now let's look at the results shown in Diagram 6. The two outermost lines (“X” and “Y”) show the object ball path and the theoretical (90 degree) cue ball path. The three parallel lines with arrows show the actual post-collision cue ball path, depending on original cue ball speed. Going from left to right, the first arrow, “a,” is the actual path for a slow-rolling cue ball, arrow “b” is the path of a moderate-speed cue ball, and arrow “c” is the path of a high-speed cue ball. In Part II of this lesson, I will discuss some of the peculiar qualities of this cue ball angle, show you how to calculate the exact angle, and show some “can’t miss” carom shots that result from the unique nature of half-ball hits with a rolling cue ball. Stay tuned. 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