The Myth of the Tangent Line – Part II

by Steve Crutchfield

 In the July/August issue of this magazine, I ended Part I of this topic with a diagram of the actual post collision cue ball paths for three (rolling) cue ball speeds.  In Diagram 1, those cue ball paths are:

(a) Slow-rolling cue ball, (b) moderate speed cue ball, and (c) fast cue ball.

 These trajectories are easy to duplicate o­n your table at home.  Place an object ball near the location shown in Diagram 1 and place a narrow strip of masking tape along the object ball path shown.  Using a builder's square or any other right-angle device you have around the house (picture frame, book, etc.), place a second strip of tape 90 degrees to the first piece along the expected cue ball path.  This second piece of tape is the tangent line.  From a distance of 4 to 5 feet, shoot the cue ball so that it makes a half-ball hit, sending the object ball down the taped path.  Hit the cue ball o­ne cue tip above center at lag speed, and with a chalk cube, mark the point where the cue ball contacts the end rail.  Place both balls o­n their original positions and repeat the process with a moderate speed, and then with a fast cue ball.  If you have a video camera, have someone record the experiment.  If you are able to make a true half-ball hit each time, you should see the cue ball follow the tangent line momentarily before it veers off at about 34 degrees from its original direction or 64 degrees from the object ball path.  The effect is especially noticeable with the high-speed cue ball.  In fact, it's more dramatic to first lay out three strips of tape at 64 degrees to the object ball path (26 degrees to the tangent line) and find the cue ball speed that causes the ball to follow the 90 to 64 degree masking tape paths.  Remember though, these are the angles for the half-ball contact. In laymen's terms, here's what going o­n.  Before the collision, the cue ball has linear momentum (or energy) because of the linear movement of its center, and angular momentum (rotational energy) because of its rolling motion.  When a ball rolls without slipping, it moves a linear distance equal to its circumference for each revolution.  Prior to object ball contact, the rotational energy of the cue ball is about 28% of its total energy.  The collision causes a drastic reduction in the linear speed of the cue ball but has little effect o­n its angular (rotational) speed.  This means the cue ball, after collision, is rotating at about the same speed but translating at a much slower speed.  In other words, its rotational energy suddenly becomes the majority of its total energy, and the cue ball is slipping o­n the cloth with forward spin.  While it is slipping (as opposed to rolling), it follows the tangent line (90 degree path).  When the friction between the ball and cloth restores rolling motion, the cue ball moves off the tangent line and follows the 64-degree path.   At very slow speeds, the 64-degree departure is nearly instantaneous, and there appears to be no tangent route at all. If you're still with me, you should have some questions.  Why 64 degrees?  Is this angle dependent o­n the cloth and its frictional characteristics?  What are the angles for collisions other than half-ball hits?  Is it ever 90 degrees?  How is this angle calculated?  Good questions.  The post collision angle between cue ball and object ball was first found by R. Even Wallace and Michael C. Schroeder, Department of Physics, University of Maine.  Their results were published in the American Journal of Physics, vol 56, in September 1988.  They found that the angle is independent of the amount of friction between ball and cloth.  More importantly, they arrived at an equation that predicts the cue ball angle for all object ball angles.  The development of that equation is really messy, so I'll just give you the results of it in Diagram 2.

 This chart shows the cue ball angle as a function of the object ball angle.  For example, when the object ball angle is zero, the cue ball angle is also zero.  This is the result of a full hit where the cue ball follows the object ball.  At the other end of the chart (an infinitely thin slice o­n the object ball), the object ball departs at 90 degrees, and the cue ball deflection angle is zero again.  This is the o­nly point o­n the curve where the total angle between cue ball and object ball is 90 degrees.  Now look at the area of the half-ball hit where the object ball departs at 30 degrees.  At 30 degrees o­n the horizontal axis, the chart reads about 34 degrees for the cue ball angle.  For the half-ball hit, the total angle is 64 degrees.  The peak of this curve actually occurs at an object ball angle of 28.2 degrees with a cue ball angle of 33.8 degrees.  Notice how flat this curve is in the vicinity of the peak.  A flat area of a curve means there is a small change in the dependent variable (cue ball angle) over a large range of the independent variable (object ball angle).  That is to say that over a limited range of object ball angles, the cue ball angle remains almost constant.  Table 1 shows just how constant the cue ball angle is for object ball angles in the range of 23 to 34 degrees.   

 For a comparison look at Table 2, which shows a similar 11-degree range (5 to 16 degrees) of object ball angles.

  In this case, the cue ball angles range from 12 to 29 degrees.  This is useful stuff.  If you want to carom the cue ball off of an object ball to tap a sitting duck o­n the edge of a pocket, the most forgiving way to do it is with a half-ball hit o­n the object ball.  This won't always be possible, but it's helpful to recognize the situation when it does exist or when you can cause it to exist.  If the half-ball hit results in the desired carom (33 degrees), and you miss the hit by 7 degrees o­n the fat side, the cue ball still takes the 33-degree angle.  Conversely, in the 5- to 16-degree range (a much fatter hit o­n the object ball) a 7-degree error is disastrous in terms of the resulting cue ball path.   Here's an example of this you can try yourself.  Place an object ball o­n the head spot, the cue ball about 2 inches left of the foot spot, and the 9 ball in the jaws as shown in Diagram 3.

 Shoot the cue ball with moderate speed and hit it a cue tip above center.  If you can come within 5 degrees of either side of a half-ball hit, the cue ball will head directly for the 9 every time.  Even a casual player can make this shot 9 out of 10 times. Another interesting way to depict the object/cue ball angles is by showing the total cue-object ball angle as a function of the object ball angle as I've done in Diagram 4.

 As you can see from the chart in Diagram 4, the total angle between cue ball and object ball increases from zero to 90 degrees as the object ball angle increases from 0 to 90 degrees. 

Again we're assuming a normal rolling cue ball condition.  Of the infinite number of points o­n this graph, there is o­nly o­ne where the cue ball follows the tangent line, and that point results in no substantial movement of the object ball.  Therefore, there is basis for the claim that the post collision angle between a rolling cue ball and a stationary object ball is almost never 90 degrees. And now you know the myth of the tangent line.

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