How should you throw the ball to increase the odds of making a strike? Many novice bowlers release the ball near the center of the lane, sending it along a straight line toward the foremost pin, the headpin. The shot is haphazard because the pin is so far away that aiming is difficult and because the action of the pins is seemingly unpredictable. Still, the play occasionally produces a strike. A seasoned bowler often adopts a more reliable strategy, in which the release is made along the side of the lane. The aim is not toward the pins but rather toward painted markers 15 feet down the lane. In addition, the ball is given a sidespin at the moment of release. The ball appears to travel parallel to the side of the lane for a short time and then to hook suddenly toward the headpin, approaching it at what seems to be a large angle. The target, which is called the pocket, is the left or the right side of the headpin.

Experienced bowlers continue to claim that putting sidespin on the ball greatly increases the chance of a strike. They may be right. The game is subject to so many variables, however, that it is hard to verify the claim experimentally. Instead I set out to test its plausibility by answering several questions theoretically. Why does a sidespin yield a curved path? Does the ball hook at some particular point along its path? Why would an angled approach to the pins increase the chance of a strike? Is the angle of approach as large as some bowlers say it is?

The bowling lane is marked by a foul line beyond which the player must not step when releasing the ball. The headpin is 60 feet from the line. The wood pins, each of which is 15 inches tall and no more than 3.6 pounds in weight, are numbered according to their position; the headpin is the 1 pin. The distance between the centers of any two adjacent pins is one foot. The horizontal cross sections of the pins are circular and their maximum diameter is less than five inches.

The ball, which may weigh up to 16 pounds and be no more than about 8.6 inches in diameter, normally has three holes into which you insert a thumb and two fingers. (Weights inside the ball compensate for the weight loss from the holes.) The surface of the ball is plastic or a hard rubber composition. To throw the ball you first swing it toward the rear and then, stepping briskly toward the foul line, swing it forward in a pendulum motion, keeping your palm below and just to the rear of the ball. As you reach the foul line you crouch, sliding on the floor with one leg extended behind you so that the ball is low. When the ball reaches the lowest part of the swing, or slightly after, you release it.

The lane, made out of narrow wood boards, is 3 1/2 feet wide, bordered on both sides by a channel. Part of the lane is treated with an oily material so that the ball initially slides along the lane rather than rolling. How much of the lane is oiled varies from one bowling establishment to the next. At my favorite place, Tuxedo Lanes on the south side of Cleveland, the first third of each lane's length is oiled, except for narrow strips along the channels.

The curved path of the ball was first investigated mathematically in 1977 by Don C. Hopkins and James D. Patterson of the South Dakota School of Mines and Technology. I shall simplify their treatment and also limit it to the case of a right-handed bowler who throws the ball directly forward on the right side of the lane. At the moment of release the fingers are brought smartly upward on the right side of the ball, giving it a counterclockwise spin. While the ball is sliding down the lane it is subject to two frictional forces. One force is rearward, opposing the forward motion. The other force, which is toward the left, opposes the spin. The rearward acceleration diminishes the ball's forward progress; the acceleration toward the left moves the ball away from the channel along which it initially slides.

Consider the ball from the perspective of the channel. If the bowler has put backspin on the ball (as some might do), the bottom of the ball has a forward speed that is greater than the speed of the center; otherwise it has the same forward speed as the center. For the ball to roll, the bottom of the ball must have a rearward speed that equals the forward speed of the center. As the ball slides, the rearward friction on the ball slows the center and also slows and then reverses the speed of the bottom of the ball. When the speeds are properly matched, rolling begins.

A similar interplay of friction and speed alters the sidespin. From a rear perspective the bottom of the ball moves rightward, whereas initially the center of the ball has no motion toward the left or the right. The friction resisting the spin slows it while also propelling the center of the ball to the left. When the speed of the bottom toward the right matches the speed of the center toward the left rolling begins. The change is simultaneous with the transition to rolling in the forward direction.

During the sliding the combined friction on the ball sends it leftward along a parabolic path whose curvature depends on the initial values of the spin rate and the forward speed. For example, a greater value for the speed or a smaller value for the spin rate decreases the curvature. At the instant rolling begins, the ball leaves the parabola along a tangent to the curve, and thereafter it travels in a straight line.

To determine the angle of this straight-line roll, which is also the ball's angle of approach to the pins, Hopkins and

Patterson estimated the forward speed and spin rate given to the ball. They also chose a representative value for the coefficient of friction, which is a measure of the surface roughness and degree of lubrication between the ball and the lane. In all their calculations the angle of approach was never more than three degrees. Such a measly angle hardly seems to justify the trouble of putting sidespin on the ball.

I wondered if the angle was small only because Hopkins and Patterson assumed that the coefficient of friction was uniformly small throughout the ball's travel to the pins. I toyed with their equations, expecting to find that the disappearance of the oily preparation of the lane about a third of the way to the pins might create a larger angle. Perhaps when the ball reached the "dry" part of the lane, the sudden increase in friction whipped the ball into a larger angle of approach.

What I found surprised me: the angle of approach is independent of the coefficient of friction. Instead it is set by the initial ratio of sidespin to forward speed. (Backspin plays a minor role.) If the sidespin is small or the forward speed is large, the angle is tiny. If the spin is large and the forward speed is moderate, the angle can be 10 degrees or even somewhat more than that.

Although the coefficient of friction does not influence the ball's angle of approach, it does determine where along the lane rolling begins. If the coefficient is large, the ball diverges from a sharply curved parabola early in its travel and so may end up to the left of the pocket. If instead the coefficient is small, the ball diverges from a mildly curved parabola late in its travel and may end up to the right of the pocket. (If the lane is fully oiled, the ball might even reach the pins before it begins to roll.) Part of the skill in bowling is knowing how to throw the ball for your lane's particular coefficient of friction. That knowledge comes from practicing on the lane before a match, adjusting your throws until the ball hits the pocket properly. The task is not easy, because with each play the ball carries oil into the dry region, altering the coefficient of friction there.

In retrospect these results are not very surprising; indeed, they are exactly in line with advice from professional bowlers. If the ball hits the pin array "low" (too far to the right) or "high" (too far to the left), you should adjust the speed and spin in the launch to change the angle of approach. Alternatively you can adjust your position along the width of the lane to move the ball's path left or right. Or you can throw the ball toward the left or the right instead of straight ahead in order to rotate the path around the launch point.

What accounts for the claims that the ball hooks suddenly? The ball can hook if it slides from an oily region of the lane to a dry one. The sudden increase in the coefficient of friction enhances the curvature of the ball's parabolic path, and the sharp increase in the ball's direction of travel toward the left is the hook. The angle of approach is not altered by the hook, however; a hook simply causes the ball to reach that angle sooner than it would otherwise.

If the angle of approach of a ball thrown with sidespin is only three degrees or so, does it actually increase the chance of a strike? To answer the question I decided to calculate what the ball does after it hits the 1 pin. I first drew an overhead view of the pin array with the aid of a template having circular cutouts. Because the center of each pin is one foot from the center of each adjacent pin, the array forms an equilateral triangle and so has 60-degree corners. Within the array the pins form smaller equilateral triangles.

I simplified the collision by assuming that it is too brief for friction between the ball and the pin to be important. (Certainly friction between the cue ball and a numbered ball is rarely important in pool.) I assumed too that the collision is elastic (no kinetic energy is lost) and that the total momentum of the objects does not change. In short, I imagined that the collision is like the collision of a large hockey puck (the ball) and a smaller puck (a pin), both pucks being on ideally slippery ice. The ball was represented by a circle having twice the diameter of a circle representing a pin; otherwise the array was drawn to scale.

Two angles are important in my analysis [see Figure 4]. One is the angle of approach by the ball. The second angle is what I call the touch angle, which is the angle from the front of the 1 pin to where the ball touches the pin, as measured around the center of the pin. At the instant of the collision the pin is propelled by a force directed along a line connecting the centers of the pin and the ball. Hence the pin travels along a path that is angled from the forward direction by the touch angle.

The collision deflects the ball from its original direction of travel, away from the direction taken by the pin. The deflection angle (which I call theta, ) depends on the angle (called phi, ) between the ball's original velocity vector and the velocity vector imparted to the pin. A curve relating and is shown below. To construct the curve I assumed that the ball is four times as massive as the pin. Note that the maximum deflection of the ball is slightly greater than 14 degrees. (I shall gladly supply a copy of the calculations for the graph.)

The collision between two pins is easier to follow. At the instant of a collision between pins the second (initially stationary) pin is propelled along a line connecting the centers of the pins. The first pin is deflected in a direction perpendicular to that line. (The only exception to such perpendicular deflection is when the collision is head on, in which case the first pin stops.)

Armed with my graph and template and a stack of photocopies of the pin array, I explored what happens when the ball strikes the 1 pin at various angles of approach and touch. If you do the same kind of analysis (either on paper, as I did, or with a home computer), keep in mind that the results are only suggestive, because they ignore many practical aspects. If a pin skips along the lane after being struck, friction can redirect it. If it lies down while moving, it can sweep a large path clear of pins. It may also fall to the lane and then spin about a vertical axis, knocking pins down. One pin might even collide with another moving pin. In addition, unavoidable drawing errors make the calculated path of a pin uncertain after it has struck more than one or two others.

To map the ball's travel through the pin array, first find its deflection after it hits the 1 pin. Extending the line of travel until the ball makes contact with the 3 pin, draw the ball at its point of collision. The pin's direction of travel is set by the line connecting the center of the pin to the center of the ball. The ball is deflected away from the pin, with its direction of travel changing by the deflection angle, . To find from the graph, measure , the angle between the path taken by the pin and the path of the ball just before the collision. Then draw the new direction of travel for the ball. The procedure is repeated each time the ball hits a pin.

Bowling textbooks describe a perfect strike as one in which the ball hits only pins 1, 3, 5 and 9. The 1 pin initiates a sequence of head-on collisions in which pins 2, 4 and 7 fall, while the 3 pin knocks down the 6 pin, which in turn knocks down the 10 pin. Once it is hit by the ball, the 5 pin drops the 8 pin. The fall of pins along the left side of the array suggests that the touch angle should be between about 20 and 40 degrees, with an ideal angle being 30 degrees, because a line along the left side lies at an angle of 30 degrees from the forward direction. With the touch angle between 20 and 40 degrees, I calculate that the 1 pin leaves the collision with as much as three times the speed at which the ball leaves it.

Concentrating on touch angles in that range, I investigated how variations of the angle of approach might alter what the ball does after it hits the 1 pin. If a nonzero angle of approach has any advantage, it should show up in such an analysis. If it does not show up, I would have to conclude either that putting sidespin on the ball is irrelevant or that (contrary to my experience in pool) the slight friction between the objects in a hard collision is somehow important.

If the angle of approach is zero, the ball deflects into the right side of the array when the touch angle is 20 or 25 degrees. If the touch angle is between 30 and 40 degrees, the deflection of the ball by the 1 and 3 pins is so severe that the ball never penetrates the array. When the touch angle is between 35 and 40 degrees, the ball slants off rather sharply-in my calculations actually leaving the array before reaching the 10 pin, which remains standing. Bowlers describe the ball as taking a strange rightward bounce off the array, as if the array were a wall. The large deflection provides a distinct disadvantage; a strike is more probable if the ball, big and massive, bullies its way right through the array.

Suppose the angle of approach is three degrees. Then at all touch angles between 20 and 40 degrees the ball does penetrate the array on the right side. Although the angle of approach is only modestly different from the preceding example, the results can be dramatically different. The advantage of sidespin leading to an angled approach is that penetration is guaranteed for any touch angle that downs the left side of the array in textbook fashion. Penetration is even more pronounced for larger angles of approach. If the angle is 10 degrees, the strike is precisely the textbook example. To throw a textbook strike, then, you should launch the ball with moderate forward speed and high sidespin, adjusting your throw to lane conditions until the ball repeatedly touches the 1 pin at an angle of about 30 degrees. Even if you err somewhat, the strong penetration of the ball into the array and the complicated pin movements I have neglected will very likely still give you a strike.

Following the ball on paper reveals another reason for not throwing the ball straight down the lane. If the ball hits the 1 pin at a touch angle of zero the 7 and 10 pins will probably remain standing as the ball plows directly through the center of the array. To knock down the 7-10 "split" with a second throw is almost impossible. Some other difficult spares are at least feasible. Consider the spare with pins 6, 7 and 10. Bowling guides state that the ball should hit on the right side of the 6 pin, sending the pin into the 7 pin; the ball goes on to take out the 10 pin. Indeed, I find that with a touch angle of about 70 degrees on the 6 pin, the 7 pin is neatly eliminated. The collision on the 6 pin actually need not be that accurate if the pin falls and sweeps a wide path as it travels across the lane.

A wealth of material remains to be explored with a pencil-and-paper approach to bowling. You might investigate the "backup ball" throw, in which the ball approaches the 1 pin high and then curves back into the left pocket between the 1 and 2 pins. There are also hundreds of spares to consider. How should each of them be played in order to knock down all the pins?

Details about the ball also raise perplexing questions. Having hit the 1 pin, the ball must slide until it begins rolling again. How does the friction on the ball during the slide change its direction of travel? (I suspect that the friction reduces the angle of deflection.) Does the nonuniform distribution of mass in the ball influence its motion? Does its large mass make it a gyroscope that resists redirection? Granted that a theoretical approach to bowling is limited because of all the variables involved in a real game, the analysis is still rewarding if it helps you to understand even approximately why the ball and the pins do what they do.

Bibliography

BOWLING FRAMES: PATHS OF A BOWLING BALL. D. C. Hopkins and J. D. Patterson in American Journal of Physics, Vol. 45, No. 3, pages 263-266; March, 1977.

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