then we have
Figure 1B Average force from the club face impact for the ball to accelerate from 0 to 150 mph in
0.00045 seconds.
to cancel out gravity, which makes for messy calculations. If we work with kilograms and meters per second, then we just plug the values straight in. So we have “force × 15 seconds” equals “1,500 kilograms × 67 meters per second”; or force = (1,500 × 67)/15, which gives 6,700 newtons of force. One pound is equal to approximately 4.5 newtons, and if you remember the tale about Newton putting all this together by watching an apple fall from a tree, then it makes sense because we typically get about 4 or 5 apples in a pound. Therefore the average force from the back wheels of your Ferrari, accelerating it forward, is 6,700/4.5 = 1,500 pounds.
Now you are on the first tee. It’s going to be a great morning, you just hit one of your best drives, and with some bounce and roll, it looks like it’s out there about 250 yards. To do this you must have hit the ball around 100 miles per hour, and it took off at approximately 150 miles per hour. For comparison with your Ferrari, it did this, from a standing start, as shown in Figure 1B, in just 0.00045 seconds!
Surprisingly, the force you applied to the ball was just about the same as the driving force produced by the wheels of your 730 hp Ferrari. In this case, we have a ball mass of 0.045 kilograms (0.1 pounds) instead of 1,500 kilograms and a time of 0.00045 seconds instead of 15 seconds. So, the calculation for force changes to “force × 0.00045” equals “0.045 × 67”; or force = (0.045 × 67)/0.00045, which gives 6,700 newtons or 1,500 pounds, exactly as before. In this case, the force starts from zero, reaches a maximum at just over 0.0002 seconds, and then decreases back to zero at 0.00045 seconds as the ball leaves contact. The maximum force is thus close to 3,000 pounds; for a better feeling of this magnitude, let’s say a “ton and a half.” So just laugh when you see those four extra miles per hour claims for “low friction” tees, which magically add 80 pounds to the maximum impact force.
We can break contact time down into two parts. At just over 0.0002 seconds, the ball was compressed to about a 1-inch diameter imprint on the face, and the clubhead had slowed down to about 82 miles per hour, with the ball traveling with it at the same speed. If you had hit a trick sticky golf ball filled with thick molasses, instead of a real golf ball with a rubber core, the process would end at this point, and the compressed ball would remain compressed and sticking to the face. At this point, the mass at the end of the shaft would be both the 0.44 pound head and the 0.1 pound ball. From Newton’s laws, it can be shown that slowed down to 82 miles per hour, the energy of the moving mass at the end of the shaft would still be 130 foot-pounds. As you get swept off your feet trying to slow down, this handful of energy, be consoled that you have just experienced zero coefficient of restitution.
Instead, a real golf ball starts to spring off the face shortly after 0.0002 seconds; the action of restitution or recovery speeds the ball up to 150 miles per hour, while the reaction slows the club head down further to 67 miles per hour. Therefore, compared to the 100 miles per hour strike against a stationary ball, at the end of the impact, we have a 150 miles per hour ball and a 67 miles per hour club head with a speed difference between the two of 83 miles per hour. So the golf ball, compared to the molasses-filled ball, has recovered 83 percent of the impact speed. In proportion terms, 0.83 of the impact speed has been recovered in speed away from the moving club face; of course, this is the coefficient of the restitution for the impact, abbreviated throughout the book as CofR. The energy of the 0.44-pound head traveling at 67 miles per hour has been halved during restitution to 65 foot-pounds, which is slowing down just provides a smooth follow-through, finishing in an elegant stance with the belt buckle facing the target—nicely done!
However, the beautiful swing would have accomplished very little without the dimples on the surface of the ball, first introduced in the early 1900s and now perfected in shape through exhaustive computer modeling and wind-tunnel testing. Using a smooth round ball, even with 0.83 CofR, your drive would probably have carried about 130 yards, with maybe bounce and roll taking it another 20 yards. We will wait until the next shot to figure that one out.
The bounce and roll must be better than you thought because you are now only 115 yards out—just made for your pitching wedge shot. Your magic golf day continues. The ball launches around 40 degrees and just seems to keep on climbing, up to somewhere around 80 feet. From its steep descent, it takes a single short bounce about 10 feet from the hole, follows a smooth spiral trajectory as it rolls off the sloped right side of the green, and has slowed down to about 1 mile per hour before crossing just inside the edge of the hole. Any faster and it would have escaped, but instead it rolls half way around the edge of the hole without touching the flagstick, runs out of steam, and topples in sideways—maybe time to just go and celebrate.
The eagle approach shot happened because you managed to put around 8,000 revolutions per minute backspin on the ball, and in so doing changed it to a little Harrier jump jet, which just kept on climbing as it appeared to do. The story of this propulsion system, with its associated low-drag performance, must wait until Chapter 2. The ball actually landed with most of this backspin still in place, which caused it to check quickly and start its slow forward roll.
One final comparison with the Ferrari, and then we will move onward. The amazing amount of backspin is produced because the ball sticks to the club face, exactly like the rubber car wheels stuck to the road in the braking turn. If we could put a microphone on the wedge, then amplify and slow down the signal, we would likely detect very high-frequency squealing as the ball is forced to rotate while gripping the face. This is exactly the same as the Ferrari tires gripping the road around the bend while different points on the area of contact must travel at different speeds. An even better comparison is the squealing of rubber-soled athletic shoes as they brake and accelerate during the step, all the while gripping the floor. Not coincidentally, the latest high-spin golf balls have covers made from the same urethane polymers as indoor running shoes. We will see later in the book how this produces such
amazing amounts of backspin that the ball actually comes off the face skidding, just as if a powerful internal spring had been wound up and then released—which is in fact what actually happens.
It’s time to describe some of the important equipment developments that have taken the game “from stick and stones” to the exciting game of today. We will restrict the discussion to approximately the last half-century. The interested reader may wish to consult Thomas (2008 and 2011) for a more detailed discussion of this topic and an enjoyable discussion of the early history of the game.