Figure 2F Relative drag of a golf ball and same-diameter smooth ball.
drag data on a single curve. This is illustrated in Figure 2F, together with a short section of the relative drag data for a smooth ball, which can be seen to be approximately twice the value as for the dimpled ball.To complete our understanding of golf ball flight, we need to know how spin rate decreases during ball flight. Spin rate is affected only by friction with the air. Unlike lift and drag, which are determined by pressure variations around the ball and are powerful forces, the braking torque applied by the air to the ball rotation is relatively small. The change of spin rate during flight is thus also relatively small. The rate of braking increases with the combined effect of both ball speed and spin rate and can be represented by the approximate relationship, which was converted to golf units from the work by Smits and Smith (1994):Spin-rate loss/second = 0.0004 × (speed) × (spin rate)For the PGA average 7-iron shot, the initial velocity and spin rate are 120 miles per hour and 7,097 revolutions per minute. The initial rate of loss of spin rate istherefore 0.0004 × 120 × 7,097 = 340 revolutions per minute per second.Using this information, and the relative drag and lift data in Figures 2E and 2F, it is relatively straightforward to construct the complete ball trajectory given the launch conditions. The procedure for doing this is described in detail later in the chapter using drag and lift data tables. In general terms, the process is carried out using the following six steps:
1. From the launch conditions determine the initial values of drag force, lift force, and rate of spin decrease. Drag and lift forces are obtained by multiplying the relative drag and lift force values by the ball cross-sectional area, the density of air, and the ball speed squared. The first two of these parameters were canceled out of the relative force calculations. The rate of spin decrease is obtained from the “spin-rate loss/second” formula.
2. Apply Newton’s laws to obtain the deceleration of the ball over a small time increment (0.001 second increments are sufficiently small) to obtain the changed ball speed; of course, include gravity as well as lift and drag.
Figure 2G Changing ball speed and spin rate values during the ball flight of the average PGA Tour player 7-iron shot.
3. Calculate the average of the ball speeds at the beginning and end of the time increment. Multiply this average speed by the time increment to determine where the ball has moved by the end of the time increment. The changed slope of the ball trajectory over the time increment emerges from these calculations.
4. Multiply the rate of spin decrease by the time increment to obtain the decreased spin rate value.
5. Determine the drag and lift forces for the changed velocity and spin rate.
6. Repeat steps 2, 3, 4, and 5 for the entire flight. Stop the process when the ball has reached the ground!
The results of carrying out this process for the average PGA 7-iron shot are shown in Figures 2G to 2I.
Figure 2G shows the changing ball speed and spin rate. The rate of decrease of spin rate and speed are largest at the beginning of the flight. Starting from 120 miles per hour, the ball speed reaches a minimum value of 40 miles per hour in 4 seconds at the top of the trajectory. It then accelerates back to approximately 55 miles per hour in the descent. From an initial backspin rate of 7,000 revolutions per minute, the final spin rate of 6,100 revolutions per minute is a loss of only 13 percent. This is the reason why it is possible to land and stop such shots on the green. If the spin rate decayed substantially during flight, then all shots would have to be played to bumpand run from in front of the green. In contrast, the average PGA professional drive launches with a spin rate of 2,700 revolutions per minute, which decays to 2,100 revolutions per minute over a flight time of 6.6 seconds.
The lift and drag forces acting on the ball during its flight are shown in Figure 2H. The black lines correspond to the PGA average 7-iron shot, and the gray lines are for the average drive on the PGA Tour. The forces in this plot are divided by the ball mass (0.1 pounds), and so the scale is equivalent to the number of “g”s acting on the ball. Note that the drag and lift forces are in line with and at right angles to the trajectory throughout the flight, as shown in the inset ball figure.
It can be seen that for the drive, the drag force on the ball starts at 2.5 times its weight and the lifting force starts at 1.6 times the weight. For the 7-iron shot,
Figure 2H Changing lift and drag forces values during the ball flight of the average PGA Tour player 7-iron shot and drive.
the drag and lift forces are almost identical through the flight and start at 1.5 times the ball weight. These are substantial forces. They are proportional to the speed squared; so as the speed drops rapidly in the first 2 seconds due to the high drag force, the drag and lift forces decrease at a proportionally higher rate.
When following the preceding steps 1 through 6, we have also calculated the positions of the ball throughout its flight. The resulting trajectories for the average PGA drive and 7-iron shot are shown in Figure 2I.
The results in Figure 2I are in very good agreement with the trajectory data published by the Trackman Company from thousands of measurements of PGA Tour players on the practice grounds. The average heights and carry distances from Trackman measurements are 31 and 267 yards for the drive and 32 and 172 yards for the 7-iron, respectively. These agreements are not surprising. In developing the model, lift and drag data from wind-tunnel testing was adjusted by the writer in a small, systematic way to provide agreement with the average PGA and LPGA Tour performance over the full range of clubs from driver to wedges. In essence, the adjustments were made to model the average behavior of the range of “premium”
Figure 2I Trajectories of the average PGA Tour player 7-iron shot and drive.
golf balls used on the PGA and LPGA Tour. These adjustments are described in detail later in the chapter.