Numinations — September, 1999

Two-Body Collisions

© 2000, by Gary D. Campbell

Let’s begin our segue this month from the subject of music. What do I know about music? I’ve spent a lot of time in bars listening to Country and Western, but this probably isn’t the best qualification. Besides, most of that time was spent with a pool cue in my hand. The “clack” of pool balls has always been a kind of music to me, even if it is only percussion. In my own pool room I usually play jazz. Music is sound. And, what is sound? It’s the collision of molecules in waves and patterns, nearly an infinite number of collisions all happening very fast. The simplest collision is a collision between two ideal spheres. And, to imagine this, pool is a pretty good analogy. Let’s see how it works.

Imagine a cue ball and an object ball at rest on a pool table. The cue ball is impelled toward the object ball and strikes it at some point. The collision can be dead center or off to one side. In pool we might assume that the balls are equal in size and weight. If they are not, they will behave differently. Let’s also say that collisions are perfectly elastic and there is no friction between the balls. This is approximately, but not exactly, what happens on a pool table.

When a cue ball strikes an object ball, the centers of both balls and the point of contact all lie on a single line. If you draw a line at right angles to this line, exactly through the point of contact between the two balls, you will have the tangent line of contact. The two most important lines in a two-body collision are the tangent line and the line of centers (both through the point of contact). Making the above assumptions (no friction, perfect elasticity, and balls equal in size and weight), the object ball will move away from the collision along the line of centers and the cue ball will move away from the collision along the tangent line. This means that their paths will always be at 90 degrees to one another. Notice that the original path of the cue ball doesn’t matter. The only effect it has is to determine how fast the two balls will be going after the collision. If the cue ball hits the object ball dead on, its path makes an angle of zero degrees with the line of centers of the two balls (they are on the same line). After collision, the cue will stop dead and the object ball will move away at the same speed the cue ball was previously moving. If the cue ball hits to the right or left of dead center, then it will depart along the right or left tangent line, respectively.

After collision, how fast will the balls be moving? A ball’s new speed will be the original speed of the cue ball times the cosine of the angle between the cue ball’s original path and either ball’s new path. To get a feel for this, consider three cases: A head on collision, a hit at forty-five degrees, and the thinnest possible cut. In a head on collision, the tangent line is at right angles to the original path. The cosine of 90 degrees is zero. So, the cue ball will not be moving at all after a head on collision. What about the object ball? In a head on collsion, its new path (along the line of centers) is identical to the cue ball’s original path. The cosine of zero degrees is one, so it picks up all of the cue ball’s original speed.

Now consider the opposite extreme, a maximum “cut” shot, where the cue ball hits the object ball at 90 degrees. Now, the tangent line is identical with the cue ball’s original path. The cosine of zero is exactly one. The cue ball continues on its original course and speed. Notice that a 90- degree cut is the same as a miss. The slightest hit at all makes the cue ball deflect slightly from its original path, so the cosine is slightly less than one, and the cue ball will depart along the tangent line at somewhat less than its original speed. The object ball departs along the line of centers. At ninety degrees its speed will be zero. At 89 degrees, its speed will be 0.0175 that of the cue ball’s original speed (the cosine of 89 degrees). But, you should realize that a cut is not with respect to the original position of the cue ball and object ball, but with respect to the point where the balls make contact. This is off to one side and must be subtracted from the maximum cut that is possible from a given position on the pool table. So, a cut of perhaps 85 degrees is about the best you can make.

A hit where the tangent line and the original path make a forty-five degree angle causes the cue ball to veer off at forty-five degrees (along the tangent line) with 0.7071 of its original speed (the cosine of 45 degrees). The object ball also departs at forty-five degrees (along the line of centers), so its new speed is the same as the cue ball’s new speed. It appears that the sum of the new speeds is more than the original speed of the cue ball. This is because total momentum must stay exactly the same. The forward speed of both balls adds up to exactly the forward speed of the cue ball, and the sideways components are in opposite directions, so they cancel out. This follows from the Pythagorean theorem. You need to draw the velocity vectors. The velocity of each ball after a 45-degree collision is 0.7071, but this is the same as half of the velocity at 0 degrees and half at 90 degrees. Drawing three lines from the point of collision, one straight on and half a unit long, one at 45 degrees and 0.7071 units long, and one at 90 degrees and half a unit long, you get two sides of a square and its diagonal. Remember, the sum of the squares of the two sides is equal to the square of the diagonal, so one-half squared plus one-half squared should equal 0.7071 squared. And, it does!

Now, let’s use our imagination even more and vary some of the assumptions. Let’s imagine that the object ball is infinite in mass. When struck by the cue ball, it doesn’t even budge. With perfect elasticity and no friction, the cue ball will bounce directly back from a head on collision. As collisions take place more and more off center, the point of contact, the centers of the two balls, and the initial path of the cue ball will no longer all be on the same line. The only factor that remains the same is that the line of centers and the tangent line are at right angles. Now, think about the initial path of the cue ball and the tangent line at contact. Think of the tangent line as a “backboard” (like a tennis or Ping-Pong backboard). It’s immovable with respect to the collision. When a ball hits a backboard, the angle of incidence equals the angle of reflection. The cue ball departs from the collision at the same angle off of the tangent line as it came in on. If it came in at ninety degrees, it will bounce straight back at ninety degrees. If it hits the tangent line at forty- five degrees it will bounce off at forty-five degrees (being, itself, deflected by exactly ninety degrees). If it grazes the object ball such that the tangent line and its original path form an angle of only ten degrees, then it will be deflected off of the tangent line at ten degrees for a total deflection from its original path of twenty degrees. All of this assumes an object ball of infinite mass.

Suppose the object ball had no mass. Now, when the cue ball contacts the tangent line it simply passes right through it undeflected. These two extremes define opposite limits. With an object ball of zero mass, the cue ball travels directly through the line of tangents. With an object ball of infinite mass, it “reflects” off the tangent line at an angle equal to the angle of incidence. So, it follows that with balls of equal mass, the angle of reflection is zero, or directly along the tangent line, exactly halfway between the two extremes. This is what I stated above, but now it becomes a bit more intuitive. Suppose we vary some more of the assumptions, but in a more realistic way with respect to the game of pool.

When you hit the cue ball, if you don’t hit it perfectly dead center, your cue stick will impart spin to it. You can hit the cue ball above, below, to the left, or to the right of center. Each of these will cause the cue ball to spin as it moves from where you strike it with the cue stick on its way to the object ball. It first skids across the table and then friction with the table imparts a forward spin to the cue ball. When a cue ball with spin hits an object ball dead on, the collision has different results that depend on its spin at the moment of impact. For example, pool players call the forward spin imparted by hitting above the center of a cue ball “follow.” This is because the cue ball follows the object ball after a head on collision. Backward spin (from hitting below the center) is called “draw” because the cue ball draws back after collision. Right and left spin are called right and left English. Maybe it’s a technique the English were the first to develop. By the way, before hitting a cue ball off center, chalk up your cue stick or you will “miscue.” Also, when you make a long shot, shoot with a little follow. When you make a short shot, especially if it’s dead on, shoot with a little draw. Both of these techniques improve accuracy by compensating for friction.

Real pool balls collide with some degree of friction. The dirtier they are, the more the friction. Friction has its most noticable effects when using English (left or right spin). When the cue ball hits an object ball dead on with left English, it deflects the object ball to the right (and the cue goes slightly to the left to conserve momentum). The opposite happens with right English. An even more noticeable thing happens when the cue ball strikes a rail (edge of the table). Without English, its angle of reflection is equal to its angle of incidence (less any spin it picks up in contacting the rail). With English you can imagine what happens. If you hit a rail dead on (at ninety degrees) with left English, the cue ball will deflect to the left. Take the same shot, but aim slightly to your right keeping the left English and the cue will rebound straight back (or even slightly to the left) instead of to the right as you would expect.

So, when you are in a bar, listening to music and shooting pool, keep in mind that even pool balls aren’t the perfect example of an ideal two-body collision. Sometimes the cue ball is larger or heavier than the object balls. Sometimes the balls aren’t perfectly clean. Friction and spin affect the way they behave. And, the ideal two-body collision, like music, is a thing of the imagination.

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