## Flight of the Golf Ball

Now we have launched the golf ball, with some ball speed, launch angle, and spin. On this page, we will see how that turns into a ball flight, with trajectory shape and distance.

This is very much affected by the air through which the golf ball flows. We all know the air does some bad things to our golf shots. It turns sidespin into slices and hooks, and air resistance -- or "drag" -- slows down the ball and steals distance. What most people don't know is that air provided a lot of that distance in the first place. In a vaccuum, the ball would not go nearly as far.

 Let's take a look at why that should be. The pictures are take from a web article on Golf Ball Aerodynamics, by Steve Aoyama of Titleist. I won't go into the aerodynamics here, but there's plenty of good articles on the Internet, starting with this one. For more, just search on [golf ball aerodynamics]; it will turn up pages of them. The first picture shows air flowing past a non-spinning ball. The air flows to the right, but that is the same aerodynamically as the ball moving to the left through the air. Think of it as a picture of the moving ball, with the camera moving along with the ball. A few ways to look at this: In terms of forces, the wind blows the ball to the right. Since the ball is moving to the left, the wind slows the ball down. The aerodynamic force slowing the ball is called "drag". In terms of energy, the air has to deflect around the ball and come back together after the ball has passed. It takes energy to move the air around like that. That energy is transferred from the moving ball to the air, so the ball slows down. Now look at the second picture. This time, the ball is spinning clockwise in the left-to-right flow of air, corresponding to backspin. The effect of the spin is to deflect the air downward as it streams past the ball. So, in addition to disturbing the airflow, the ball is pressing the air down. What would Newton have to say about this? "Equal and opposite reaction." The ball exerts a force on the air to push it down. The reaction is an equal force that the air applies to the ball, pushing it up. This force, shown as a red arrow in the picture, is called "lift". The lift force has a magnitude and a direction. The direction is always at right angles to the direction the ball is moving, and is in the plane that the ball is spinning.The magnitude increases with ball speed, with the spin, with the density of the air, and depends on some aerodynamic properties of the ball's surface -- like the dimples. The result is a collection of forces as the ball moves through the air. The ball's motion is described by its speed, direction, and spin at any point along its path. The forces are: Drag, in exactly the opposite direction to the one the ball is moving. Weight, straight down. Lift, perpendicular to the path of the ball and in the plane of the spin. Note that I did not say that lift is an upwards force. It is at right angles to the path of the ball, and in the plane of spin. Here are a couple of interesting results of that distinction: The spin plane doesn't have to be straight up and down. In this picture, the spin plane is inclined a little to the left, so the lift force (black arrow) is also inclined to the left. Therefore, it has components (gray arrows) upward and leftward. Yes, lift still produces an upward force. But it also has a force to the left. This will be a centripetal force, which will curve the flight of the ball to the left... a hook for a right-handed golfer. So the spin on the ball is a combination of mostly backspin but some sidespin as well, and that means that the resulting lift force is a combination of upwards force keeping the ball in the air and hook or slice force curving the ball left or right. During the time that the ball is climbing, the path of the ball is tilted upward. Since the lift force has to be perpendicular to the path of the ball, it is tilted "backward". That is, the lift has an upward component keeping the ball in the air, and a backward component decelerating the ball from it's progress down the fairway. So the ball slows down during its climb, not just from drag but also due to the backward tilt of the lift.

### From force to trajectory

So air is our enemy (drag) but also our friend (lift). Lift keeps the ball in the air longer, giving us more distance. Which brings us to a question I'm asked all the time: is there some simple formula for distance? The answer is that there are several, but none are very good. In particular, any simple formula that includes loft is too simplistic to reflect the complex actions of aerodynamics, especially for the longer clubs.

The way trajectories are calculated are not by plugging numbers into a simple formula. Rather, a computer program is used to compute the forces on the ball and, a few inches or feet at a time, move the ball through its trajectory. Such programs aren't limited to the club designer's (or clubfitter's) personal computer; the same kind of program is used in launch monitors and golf simulators.

I wasn't planning on going into how trajectory programs work. But then, I realized that their working gives insight into how trajectories themselves work -- how the physics turns the forces into a trajectory. So here goes; here's how those programs work:
1. Start at the instant of launch, knowing the launch conditions.
2. The program starts with the speed, direction, and spin of the ball at that instant. From that the program can figure -- or already knows:
• The three forces on the ball: lift, drag, and weight.
• The acceleration of the ball, just by applying F=ma. This is acceleration in three dimensions, so it includes up-down and left-right acceleration, as well as downrange acceleration.
3. The program steps forward a tiny increment in time, probably something between 1/100 and 1/10 of a second. That involves:
• Moving the ball in space, in the direction of the current velocity. For instance, if the velocity is 200 feet per second and the interval is 1/100 of a second, then the ball moves 2 feet in the direction the ball was already headed. (That's 200 feet per second times 1/100 of a second.)
• Computing a new velocity. We have an old velocity, an acceleration (which may not be in the same direction in the old velocity -- but we know how to handle that), and a time interval over which the acceleration acts. So it's easy to compute the new velocity. The new velocity consists of a new speed and a new direction, both slightly different from the old ones.
• Computing a new spin. The spin just loses a small percentage of its magnitude every time interval, due to air resistance.
4. The program now has a new set of conditions: speed, direction, spin -- and a new position. Go back to step #2 with the new information and repeat. Continue this loop until the ball's height is zero; that means it has hit the ground.
The remarkable thing is that this is how the real world operates, too. It does so continuously, not a step at a time. But it works by continually looking at the velocity, turning that into forces using F=ma, and changing that velocity according to the acceleration.

### Spin, lift, and ballooning

As we said, lift is our friend. Right? Well, not always. There can be too much of a good thing. For instance, you've heard about a ball losing distance because it "balloons". That is what comes from an overabundance of lift. Let's see this in more detail. (The studies below were done using trajectory programs from Max Dupilka and Tom Wishon. I needed both programs, because each is better than the other at some job.)

### Numbers and the "Launch Space"

Now we know that the price of ballooning changes with clubhead speed. Let's look at this a little more closely, since it is so important to clubfitting -- especially drivers. Today, the state-of-the-art method of fitting drivers is with launch monitors, and the thing the launch monitors look at are, of course, the launch conditions: ball speed, launch angle, and spin. How do these relate, when the objective is maximum carry distance?

Engineers and mathematicians would call this the "launch space". It is a graph that starts with the launch parameters and plots the carry distance. That's a four-dimensional space (ball speed, launch angle, spin, and distance), which is hard to show on a two-dimensional page. But let's see if we can find some useful ways to visualize it.

Here's how carry distance varies with spin for several representative ball speeds. For each speed we used a good launch angle for that speed and just varied the spin. Points to note:
• The four ball speeds used were:
• 100mph - a senior or less athletic golfer, with a clubhead speed about 70mph.
• 130mph - this represents good impact at 85-90mph. It corresponds to the majority of decent golfers.
• 160mph - a big hitter that you might run across at your course.
• 190mph - one of the bigger hitters on the pro tour, but not yet a serious competitor in long drive championships.
• The best spin for the 190mph golfer was about 2000rpm, while the 100mph golfer did best above 3000rpm. Actually, the optimum was probably around 4000rpm, but the distance was very close to that at 3000rpm.
• For the 190mph golfer, that 2000rpm was with a launch angle of 10°, while the 100mph golfer had a launch angle of 20° and did best over 300rpm. So the effect of spin is even more than it appears. That's because the lower launch angle needs more spin to keep the ball in the air.

Now let's look at the other side of the picture: launch angle. Here are the corresponding plots, using a good spin for each speed and plotting distance against launch angle.
• The golfer with the highest ball speed does best with the lowest launch angle: 190mph at 11°. The best launch angle for a golfer with only 100mph ball speed is more than double that, at 23°.
• As before, we held the spin constant at a "good" value for each ball speed. That spin varied from 2000rpm for the 190mph golfer to 3500rpm for the 100mph golfer.
One of the most startling things on this page is how high the optimum launch angle is for most golfers. Next we'll see what's really going on.

Now we've seen how distance varies with spin, and how it varies with launch angle. But it's very instructive to see how distance varies with both at the same time. Let's add a dimension to what we're looking at. Here's a chart that shows more of the launch space -- distance vs both spin and launch angle for a ball speed of 124mph. (That corresponds to a well-struck ball off a driver whose head speed is 85mph, comparable to a typical male golfer.)

 Spin(rpm) Launch Angle (Degrees) 8 10 12 14 16 18 20 22 24 26 5500 193 195 196 196 195 194 192 190 187 183 5000 193 195 197 197 197 196 194 192 189 185 4500 192 195 197 198 198 197 196 194 191 188 4000 191 195 197 199 199 199 198 196 193 190 3500 190 194 197 199 200 200 199 198 195 193 3000 187 192 196 198 200 200 200 199 197 195 2500 183 189 194 197 199 200 201 200 199 197 2000 178 185 190 195 197 199 200 200 200 198 1500 170 179 185 190 194 197 199 199 200 199 1000 159 168 177 183 188 192 195 197 198 198

Because it is difficult to make much sense of a table of numbers, I have color-coded it to give it some contour -- so the colors are the third dimension. The colors go from hot (red) at the maximum distance of 201 yards to cool (blue) for all distances under 194 yards. While the location of the peak is pretty clear, the shape of the space is interesting. The peak appears to be on a "ridge" of good yardage that runs from upper left (low launch, high spin) to lower right (high launch, low spin).

The smaller, annotated picture at the right shows where the ridge is. Distance falls off rapidly on either side of the ridge. At the lower left -- low launch and low spin -- distance is lost because there isn't enough aerodynamic lift to keep the ball in the air. At the upper right -- high launch and high spin -- ballooning kills the distance. But moving along the ridge, you can change launch angle and spin by pretty large amounts with fairly little loss of distance.

"Moving along the ridge" means you have to change both angle and spin so you stay on the ridge. Let's see what happens if you just change one or the other. Suppose you had a driver that launched the ball at 124mph (this whole table assumes a ball speed of 124mph), with a 14º launch angle and a 4000rpm spin. The resulting distance is 199 yards. The peak distance of 201 yards lives at 20º and 2500rpm.
• Suppose you changed the launch angle to 20º without changing the spin. You would lose a yard, to 198 yards.
• Suppose you changed the spin to 2500rpm without changing the launch angle. You would lose two yards, to 197 yards.
But, if you change both, you'll pick up the two yards to the theoretical maximum.
This is very significant in the fitting of drivers. Here is a table showing a "family" of drivers. The design is very simple: 200g clubhead, a COR of 0.83 (the maximum allowed by the USGA), nothing fancy about the weight distribution, etc. Each row of the table shows a different loft; that is all that is varied from one driver to another in the "family". (Actually, it is effective loft at impact, a number which includes the effect of shaft bend.)

What the table shows for each driver is the distance, launch angle, and spin for each driver in the family -- assuming the ball is being struck at an 85mph clubhead speed and a zero angle of attack.
Because we have a distance, LA, and spin for each driver, we can plot the drivers on the launch space table we just calculated. When we do this, we see them lying along the red dotted line. A few points worth noting here:
• Increasing loft increases both the launch angle and the spin. This means that it does not move along the ridge, but almost at right angles to it.
• Since it crosses the ridge at a right angle, there is a fairly well-defined maximum distance as you vary the loft. And the maximum along the red line of loft variation is where it intersects the black line of the ridge. This should not be a big surprise when you think about the shape of the contours on the chart.
• The maximum distance along the red line is a long way from the peak of the launch space. The difference is 6º of launch angle (14º to 20º), and 1500rpm of spin (4000rpm to 2500rpm). Even so, the distance that driver can hit the ball is only two yards short of the theoretical maximum for that ball speed.
This last point is important! It tells us that a practical, well-fit driver is one that lives at the top of the ridge. It doesn't have to be at the peak, but it can still give almost as much distance as the peak. I've plotted the same chart for clubhead and ball speeds from long-drive champions to little old ladies from Pasadena. The simple driver that lives on top of the ridge is never more than 2 yards short of the theoretical maximum.