My favorite subject was always math.
—Tiger Woods
Tiger likes math. So does Phil. The website for the Mickelson ExxonMobil Teachers Academy quotes Phil: “As someone who uses math and science every day in my career, I recognize the importance of encouraging children’s interests in math and science and equipping educators with the tools and resources they need to succeed in the classroom.”1
This is not to say that Phil and Tiger prepare for the final round of a major tournament by solving math problems. However, it is reasonable to say that an important component of their excellence is an ongoing analysis of their swings and results. This analysis relies heavily on basic concepts from mathematics, physics, and other disciplines.
In this book, we look at various aspects of mathematics which can shed light on some of golf’s most perplexing questions: Why do we miss so many putts? Why are the new drivers so big? Does the saying “drive for show and putt for dough” apply to the modern game? My hope is that these mathematical morsels will help you enjoy the game more.
We will start by examining the geometry of golf, where the mathematics is often inextricably bound to physics. A number of excellent books on the physics of golf already exist.2 We will borrow from them frequently as we proceed through this chapter.
Sir Isaac Newton (1642–1727) constructed a framework for the analysis of the motion of all things, both on and off Earth. There is no evidence that Newton himself played golf, but by the 1600s the game was already flourishing in Scotland and England.3 Newton’s second law of motion, fundamental to our understanding of the flight of a golf ball, is often expressed by the famous formula
F = ma.
In this amazing equation, F stands for the sum of all forces acting on an object (such as a golf ball), m represents the mass of that object, and a represents its acceleration. Acceleration, in turn, is the rate of change of velocity, and velocity gives both the speed and the direction of motion of the object. In theory, then, knowing all of the forces acting on an object lets us compute its acceleration, from which we can recover the velocity and position of the object. This idea works well enough to safely land astronauts at precise locations on the moon.
The force F and acceleration a are in boldface to indicate that they are vectors. This means that they have both size (magnitude) and direction. Newton’s equation tells us that acceleration occurs in the same direction as does net force and, further, that the magnitude of the force equals mass times the magnitude of acceleration.
There are three forces that are dominant for the motion of a golf ball in flight: gravity, air drag, and the Magnus force. Gravity is the most familiar of these. In everyday language, “weight” refers to the magnitude of the force due to gravity. The weight of a golf ball is approximately 1.6 ounces and varies slightly depending on altitude and latitude. The mile-high altitude in Denver reduces weight by about 0.05% compared to sea level, while the bulging of the earth at the equator causes a weight reduction of about 0.3% moving from 50° latitude to the equator. The increased distance for drives in high-altitude places like Denver is primarily due to a decrease in air drag,4 our second force.
Air drag is what most people think about when they hear the phrase “air resistance.” It is a force that directly opposes motion, and its magnitude depends on the speed of the object. You have experienced this when you hang an arm out of the window of a moving car. You feel more air drag at higher speeds. The air drag on a golf ball depends on numerous factors, including air density5 and the dimple patterns on the ball.
The third force, the Magnus force, affects every drive, sometimes to our embarrassment. Caused by the spinning of the ball, the Magnus force acts at right angles to both the direction of motion and the spin axis and can produce wicked hooks and slices.6 The dominant type of spin in golf is backspin; figure 1.1 shows the directions of the three forces on a ball hit with backspin.
Imagine that the ball in figure 1.2 is moving into the page with backspin. As shown in figure 1.2a, the Magnus force is upward. If the spin is not pure backspin, but has some sidespin mixed in, then the entire figure tilts and the Magnus force has a sideways component. In figure 1.2b, the force is up and to the right; if you are right-handed, you just hit an ugly slice. Figure 1.2c shows a Magnus force that is up and to the left, creating a hook for a righthander.
Figure 1.1 Directions of the primary forces on a golf ball with backspin
Figure 1.2 (a) The Magnus force is perpendicular to the spin axis; (b) a slice spin; (c) a hook spin
The usefulness of dimples on a golf ball can be explained by the drag and Magnus forces. For a golfer, drag is bad. It slows the ball down and decreases distance. On the other hand, a Magnus force is generally good. Backspin is the dominant spin on a golf shot, and the resulting Magnus force has an upward component. This causes the ball to go higher, stay in the air longer, and (usually) have time to travel farther. Figure 1.3a shows a contrast between the drag forces on a smooth golf ball and on one with dimples. Notice that the magnitude of the air drag on the dimpled ball is much smaller, especially at high speeds; thus, the dimpled ball travels much farther. In figure 1.3b, the upward component of the Magnus force (or lift force) is shown for smooth and dimpled balls.7 This time, the magnitude of the force for the dimpled ball is much larger. Again, this is good. The dimpled ball goes higher, travels farther, and is easier to control.
Figure 1.3 (a) Drag force on smooth and dimpled golf balls; (b) lift force on smooth and dimpled golf balls. Used with permission of Acushnet Company.
New dimple designs are tested in wind tunnels or simulated on a computer to see if better drag and Magnus profiles are produced.
You may have a vague memory from high school that the path of a ball in flight is a parabola. Like many myths recalled from our innocent youth, there is an element of truth here, but the full story is much more interesting. Parabolas are the paths of objects on which gravity is the only force.8 This ignores air drag and the Magnus force and, therefore, is grossly inaccurate for describing most flying objects on Earth.
So why do we study parabolas in school? In a word, convenience. The “gravity-only” model produces nice equations and allows us to calculate a variety of quantities. We trade off some accuracy to keep the mathematical difficulty under control. How much accuracy do we lose? Figure 1.4 shows trajectories for two drives with launch conditions (234 ft/s is about 160 mph, typical for a professional drive) that are identical except that one ball is acted on only by gravity and the other ball also experiences air drag.9 The most obvious trajectory change is that the ball acted on by drag and gravity flies about three-fourths as high and less than two-thirds as far. Also, drag ruins the perfect symmetry of the parabola. The part of the curve to the right of its peak is shorter than the part to the left of the peak. This is due to the air drag continually reducing the speed of the ball. (In fact, at the end of the flight, the speed has been reduced to about 123 ft/s, just over half the initial speed.)
Figure 1.4 The effect of drag on a drive with initial speed 234 ft/s (160 mph) and initial angle 15°
Figure 1.5 The effect of drag and Magnus force on a drive with initial speed 234 ft/s (160 mph) and initial angle 15°
We can also add in a term for the Magnus force.10 The resulting path is overlaid onto the original gravity-only graph in figure 1.5. Notice that the Magnus force propels the ball three times as high and almost as far as in the gravity-only graph, restoring most of the distance lost to air drag.11 Carefully examine the shape of the trajectory. It starts out nearly linear, rises to a peak, and drops down, approaching a straight line at the end. This should match your experience on the golf course. Notice that the peak of the trajectory is reached about 60% of the way to the landing point. This lopsided path must be taken into account when trying to clear a tree.
As illustrated in figure 1.2 the vertical component of the Magnus force on a golf ball hit with backspin is upward. This is the source of the extra height seen in figure 1.5. An interesting consequence of this upward force is that the ball lands softly. Even without taking into account the effect of spin on a ball nestling into a lush green, this can be seen by noting that the landing speed of the ball acted upon by gravity, drag, and Magnus force is about 100 ft/s, less than half of the landing speed of the gravity-only ball.
The most dramatic consequences of the Magnus force on a golf course are hooks and slices, which are caused by sidespin. As seen in figure 1.2b, and 1.2c, a ball spinning in a plane that is not vertical will curve to the side. We’ll look at some examples. To be clear, for a right-handed golfer, a “slice” is a shot that curves from left to right; a “hook” curves from right to left; a “push” has initial direction to the right of the target; and a “pull” starts to the left of target. For a left-handed golfer, all of these directions are reversed.
The golf swing is all about delivering the clubhead to the ball in good form. Jim Furyk’s loop, Bubba Watson’s extension, Sergio Garcia’s lag, and all of the endlessly analyzed spine angles and head bobs serve to determine the clubhead speed, direction of motion of the clubhead, and the direction in which the clubface is pointed. A mismatch between the last two creates spin. In particular, if the clubhead is not moving in the same direction as the clubface is pointing, you are likely to see a hook or a slice. Therefore, swing angle and face angle are critical. In the following discussion, the vertical motion of the ball is ignored, and we are solely interested in an overhead view of the ball’s flight. The next several figures illustrate what happens if one of these angles is non-zero.
First, suppose that at impact the clubhead is moving exactly on line with the target but that the clubface is open by 5°. For a right-handed golfer, this means that the clubface points 5° to the right of the target. As shown in figure 1.6, the ball is launched to the right about 4° off-line. As seen in figure 1.7, the ball then slices well to the right and lands 260 yards down range but over 65 yards to the right of the target. The overhead view points out an interesting optical illusion. By the end of its flight (to the left in the figure), the ball is moving substantially off-line. In fact, at the halfway mark of the trajectory, the ball has moved only about 20 yards off-line, but by the time the ball lands, it is over 65 yards off target. In spite of this, the ball does not appear to be curving much at the end of its flight. As the ball slices, its velocity starts to line up with the Magnus force; because the Magnus force is reduced, the ball curves less. This, of course, is of little comfort to the golfer whose drive has already disappeared into the woods.
Figure 1.6 An overhead view of initial direction of ball from an open face
Figure 1.7 An overhead view of push/slice (ball moves right to left)
Let’s look at what happens when the clubface is aligned to the target but the swing plane is not. In particular, suppose that at impact the path of the club is 5° to the left, as shown in figure 1.8. The ball is launched slightly to the left (about 1°) of target. For a right-handed golfer, this is an outside-in swing that produces a slice. As seen in figure 1.9, the ball starts off slightly to the left and then slices back to the right, landing 264 yards down range and 44 yards to the right of the target.
If both the swing angle and the face angle are 5° to the left, the result is simply a pull (for a right-hander). Because the swing and the clubface have the same alignment, there is no slice or hook. The ball lands 278 yards down range and 24 yards to the left of the target. This drive is longer than the previous two, showing that a slice typically costs you a little distance.
The geometry in figures 1.6 through 1.9 holds in general. That is, if the face angle and swing plane are different, the ball will curve in the direction from the swing plane to the face angle. The initial direction of the ball is between the face angle and swing plane and is primarily determined by the face angle. More precisely, the initial direction is about 80% of the way from the swing plane to the face angle.12
Figure 1.8 An overhead view of initial direction with swing plane to the left
Figure 1.9 An overhead view of pull/slice (ball moves from right to left)
In the context of this analysis, a hook is simply a slice reversed. That is, if the swing angle is on target and the club face is pointed 5° to the left, the ball trajectory will be identical to that of figure 1.7 except it will curve to the left. The landing point will have the same down-range distance of 260 yards and will again be off-line by over 65 yards (but this time to the left). The common folklore that hooks go farther than slices is based on other factors. Hooks often have a lower trajectory that creates more roll.13
These calculations collectively illustrate one aspect of what teaching professionals mean when they say that the ball flight will tell you what your swing is doing. As we have seen, the initial direction of motion of the ball is primarily determined by the face angle. A pull to the left means that, at impact, your clubface is pointing to the left. Slice and hook tell you how the swing plane relates to the clubface. A slice to the right tells you that your clubface is pointed farther to the right (or, possibly, less to the left) than your swing angle. A good teacher can take this knowledge and reverse-engineer the ball flight to identify specific aspects of your swing that need attention.
The 18th hole at Hanging Rock Golf Club in Salem, Virginia, is a par 3 with a significant drop of over 40 feet from tee to green. My golfing buddies and I sometimes argue about how much difference the drop makes. This is our next topic.
The first surprise for experienced golfers is how small the angles in golf actually are.* For example, imagine a par 3 of 160 yards with a 40-foot drop in elevation from tee to green. For comparison purposes, the 40-foot drop is equivalent to the height of a three-or four-story building. How steep is the slope? It will look like an impressive drop, but some triangle trigonometry in figure 1.10 shows that the angle is tan−1 (40/480), which is about 4.8°.
Four and five degrees may not sound like large angles, but they represent very steep slopes on a golf course.
How much effect does sloping terrain have on an approach shot? Figure 1.11 shows a possible trajectory, with the ball landing on level ground at the 160-yard mark. Superimposed on this graph is a line representing a downhill slope, with a drop of 40 feet over the 160-yard horizontal distance. The intersection of the ball path with this line shows that the ball would hit the ground at nearly 170 yards, about 10 yards beyond the 160-yard mark.
As you can see, the slope creates a significant change in the distance from launch point to landing point. The downhill shot goes about a club too long. This is quantified in table 1.1.14
Figure 1.11 The effect of a downhill slope on distance
If you ask golfers, you will hear different rules of thumb for judging the effects of slopes. A one-club (10 yards) difference for every 10 yards of elevation is a simple, rounded-off version. In table 1.1, a 30-foot change in elevation increases the distance by 8.4 yards going downhill and decreases distance by 9.1 yards going uphill. Rounding up to 10 yards simplifies the rule while keeping it reasonably accurate. However, there are many factors not accounted for in the calculations.
One factor that can be varied is the target distance. Using the same parameters as in figure 1.11,15 we see that a 140-yard iron shot varies 7 to 8 yards with 10 yards of elevation change. A 180-yard iron shot varies 10 to 11 yards with a 10-yard elevation change. The longer the shot, the more the elevation shift affects the distance, even without taking into account differences in how far the ball rolls.
Table 1.1 Carry distances for different ground slopes
Figure 1.12 Two uphill shots: the higher trajectory hits the ground closer to the 160-yard target than does the lower trajectory
Another factor is the shape of the shot. Figure 1.12 shows two 160-yard iron shots, one the same as in figure 1.11 and the other with a higher loft.16 The two ball paths intersect the horizontal axis at the same point, indicating that on level terrain they would carry the same distance. However, on the uphill slope, the steeper trajectory intersects the ground line farther to the right than does the flatter trajectory. This shows that the uphill slope reduces distance less on the high-trajectory ball than on the low-trajectory ball. The high-trajectory ball lands closer to the 160-yard target than does the low-trajectory ball. A similar result holds for downhill slopes.
Numerically, table 1.1 shows 8 to 9 yards of difference for slopes of 30 feet. For the more lofted flight in figure 1.12, the difference is 7 yards.17 This is another popular rule of thumb: 30 feet of slope changes distance by 7 yards. The rule varies, depending on how high one hits the ball.
Another important factor for distance adjustments for irons is the orientation of the green. Werner and Greig have shown that the amount of bounce and roll after the ball lands depends critically on the tilt of the green. For flat (untilted) greens, they recommend one club for every 20 yards of elevation.18 For greens that are tilted toward the golfer, as most are, the roll distances change and the calculations shown above are reasonably accurate.
In the past, knowing the adjustment rule was like knowing the swing weight of Tiger’s clubs. While it might be interesting, it would not help your game. Standing on the tee, unless you could judge whether a drop in elevation was 20 feet or 40 feet, all of the rules were useless. Nowadays, however, some range-finders display adjustments for slope. If you use such a device, pay close attention to its accuracy and determine if the range-finder’s rule is based on the shape of somebody else’s shots. You may need to adjust the adjustment.
We close the chapter with a detailed look at the angle of flight of the ball. In particular, at what angle does the ball hit the ground? While this can vary from shot to shot, we can gain important information from an equilibrium calculation.
Equilibrium is a basic property of any system that mathematicians and engineers explore. An equilibrium is a balance point where the variables that are being tracked do not change. Equilibrium values may also serve the role of being the natural values that occur in the absence of external forces. For example, an equilibrium value for a bowl is the bottom of the bowl. If a ball starts somewhere in the bowl, a reasonable prediction for where the ball will be a minute later is at the bottom of the bowl. Unless someone flicks the ball or the bowl is turned upside down, the natural position of the ball is at the bottom. Some greens have “collection points” that operate this way. A ball that starts near the collection point will end up at the collection point.19 What about the equilibrium of the angle of flight?
Returning to the downhill par 3 shown in the photo of the 18th hole at Hanging Rock, on several occasions I have been in a group that disagreed about how much difference the drop makes. One golfer says two clubs, another thinks between one and two clubs, and I usually claim one club. Consistent with the calculations shown earlier in this chapter, the three actors in this drama have different ball flights. The two-club advocate hits a flat ball, I have the highest trajectory, and the one- to two-club advocate is in between. So we could all be correct.
Mathematicians look for patterns. Inventing and solving little mental puzzles is the modus operandi of the mathematical mind. An interesting thought experiment here is to imagine a golfer who hits irons much higher than I do. Based on the pattern discussed above, this golfer would need to adjust less than a full club on our downhill par 3. Now, extrapolate this to a mythical golfer who hits the ball almost straight up. Does the mythical golfer have to make any adjustment? Stated differently, is there a trajectory so steep that when you superimpose it on figure 1.11 there is essentially no difference in horizontal distances?
The answer is no. The reason is backspin. The right angle formed by the direction of motion and the Magnus force is maintained throughout the flight of the ball. If the ball is on the way up, as in figure 1.1, the Magnus force from backspin points up and back. If the ball is coming down, the Magnus force is up and forward. Imagine rotating the direction of motion in figure 1.1 by 60° clockwise. The Magnus force and air drag directions also rotate by the same angle, so that the Magnus force has a forward component (to the right in the figure). If the ball is moving straight down with backspin, the Magnus force is forward. When the Magnus force pushes it forward, the ball is no longer moving straight down. So, if there is backspin, the ball cannot literally drop straight down, but can it maintain a near-vertical drop? This is an equilibrium question.
If there is an angle that the trajectory maintains on the way down, then that angle corresponds to an equilibrium value for the slope of the trajectory. In figure 1.11, it does look like the graph has straightened into a line with constant slope.
An equilibrium value for the slope of the ball trajectory can be found by solving an equation obtained by setting the sum of the forces on the ball equal to zero.20 Actually, the equation is ugly enough that we can only estimate the solution. For a constant spin rate of 5,000 rpm, the equilibrium slope is approximately −0.92. This makes an angle of about 43° with the horizontal, which is not even close to being vertical! By comparison, a constant spin rate of 3,000 rpm has an equilibrium at 47° from the horizontal. This is steeper than 43° but still far from the 90° vertical mark.
Do golf balls actually reach these equilibrium values? Thinking that 43° looked right for figure 1.11, I created figure 1.13 confident that the slope would slide down to −0.92 (43°) and level off. This figure shows the slopes of a trajectory for an iron shot hit from an elevation of 50 feet with a spin rate of 5,000 rpm. The lower horizontal line shows the equilibrium value of −0.92. Instead of the slopes leveling off at the equilibrium, they just blow right through what I expected to be a barrier! Fortunately, the computer allows experiments corresponding to ridiculous situations. Figure 1.14 shows the slopes for a trajectory of a shot from an elevation of 2,000 feet.
Figure 1.13 The slopes of the trajectory of an iron hit from an elevation of 50 feet, passing through the equilibrium value of −0.92
Figure 1.14 The slopes of the trajectory of an iron hit from an elevation of 2,000 feet, eventually approaching the equilibrium value of −0.92
Given long enough to fall, the slopes do approach the equilibrium value. The result applies only to a hole you would see on an “Impossible Golf Holes” calendar, but the math works.21
As a final comment on this investigation, I should confess that the assumption that the spin rate remains constant is not exactly valid. Although the actual rate at which the spin decreases is not agreed upon (see note 10), if the spin rate changes by any reasonable amount, there is no single equilibrium value to be approached.
The theoretical lesson to take from this is that for a typical iron shot, whether you hit the ball high or not, the ball drops onto the green at an angle reasonably close to 45°. The practical lesson is somewhat different. Small differences in angles due to swing mechanics or equipment make a large difference in the distance carried and in the amount of roll. The higher the trajectory, the less difference the slope of the ground makes. The adjustment that you should make depends on your ball flight and may be different from the adjustments of your playing partners.
