



Article Contents 
A computer study
Such an interesting question probably calls for another pass at an
answer. Not many had access to a computer in 1968; just about everybody
has one today. I used two computer programs: 
Clubhead
mass (Grams) 
Clubhead
speed (MPH) 
Ball
speed (MPH) 
Carry (Yards) 
100 
121.7 
150.3 
240 
110 
119.0 
151.0 
241 
120 
116.2 
151.1 
242 
130 
113.8 
151.1 
243 
140 
111.5 
150.8 
242 
150 
109.2 
150.1 
241 
160  107.2  149.3  240 
170  105.1  148.3  238 
180  103.2  147.1  236 
190  101.4  146.1  234 
200  99.6  144.8  232 
210  98.1  144.0  230 
220  96.3  142.5  227 
230  94.8  140.8  224 
240  93.4  139.8  222 
ComparisonA look at the tables  or, to see more easily, the graph on the right  shows that the models do not give the same result. Not even close.The most apparent discrepancy, which can be seen in the tables, is that the total driving distance is not the same at all. The drives in Cochran and Stobbs are much shorter. However, that is easily explained and compensated for. Cochran and Stobbs did their research in the 1960s, well before spring faces extended the Coefficient of Restitution to 0.83. It is generally agreed that a driver with no spring effect will give a COR of about 0.78. If you apply that COR to the computer model, you get about 223 yards, plenty close enough to the C&S data.^{[2]} In the graph, we have scaled the C&S data as if it were gathered with a more modern driver and ball. We chose the scaling factor so the two would be the same distance at the conventional driver head weight of 200 grams. Even then, the models give drastically different results! The maximum ball speed and distance occur at a much lower head weight: around 130g rather than C&S' 200g. On top of that, the C&S maximum is much broader and the computer simulation much sharper. The shape of the two curves is very different, not just the scale. Something is wrong here. Can we get any more information that might shed some light on the difference? 

Daish's studyRod White (author of the article on Golf Swing Physics) brought my attention to a similar study by C. B. Daish in his classic book "The Physics of Ball Games". Daish's approach was to measure the variation of clubhead speed vs clubhead weight, using four human test subjects who spanned a variety of phsyical strength and golfing talent.Here are the curves for the four golfers individually. The vertical axis is clubhead speed and the horizontal axis clubhead mass, but both axes are a logarithmic scale. In playing with the data, Daish had seen that the curves would be straight lines on a loglog plot like this, which gives some insight into how we can express things mathematically. In particular, a negativesloped straight line on a loglog plot can be expressed as: M_{head} V_{head}^{n} = constant
The slopes are similar  almost the same for three of the four golfers, and not very different for the fourth  at an average value for n of 5.3. So the equation for clubhead speed vs clubhead mass is: M_{head} V_{head5.3} = constant
What is the value of constant? Well, it stays constant for any golfer as the head weight varies, but its value depends on the strength and skill of the golfer. A bigger hitter has a higher value for constant. By doing some algebra on this, we can solve for ball speed as a function of clubhead mass.
where K is a constant that depends on the golfer's strength and the clubhead's coefficient of restitution, and m is the mass of the ball. In the graph at the right, we use this formula to add the Daish model to the two we had before, Cochran & Stobbs and the computer simulation. I selected K to give a carry distance of 232 yards at 200 grams of head weight, the same as the other two studies at this reference value. The Daish study looks a lot more like the C&S study than the computer simulation. Like the C&S study, Daish has a very broad "peak" at about 200g. Since we know that Daish measured clubhead speeds for real golfers, and they all looked pretty consistent, it is probably the most trustworthy answer. But why is it so different from the computer model? Here are some thoughts on the difference:


Torquevelocity computer studyWe will use the torquevelocity curves from Sasho MacKenzie's model of the golf swing. That model is:
The omegas are rotational velocities, the Ts are torques, and Γ is a "form factor". We use a doublependulum model, and the torque is completely shoulder torque (which is torso and shoulder rotation in the MacKenzie model). For torso and shoulder rotation, ω_{max}=30radians/sec and Γ=3. So the torque reduction becomes:
The graph to the right shows what the torque reduction looks like. It is a pretty serious reduction. The torque is reduced by half at 6 radians/sec; that's the same as 344 degrees/sec. Torso/shoulder rotation during the swing can easily be twice that, so the torquevelocity reduction is very significant. I do not have the tools at hand to do the simulation exactly, but I can certainly do a backoftheenvelope study to see if the effect is the right order of magnitude to explain the difference between the studies. The graph at the left is produced by SwingPerfect, and is the angular velocity of the arms' rotation during the downswing. For all the swings of interest, this curve is roughly the same shape; only the scale changes. That means we can get a firstorder approximation by picking a representative velocity and using that in our torquevelocity curve. Since we get the biggest effect on torque at high velocities, and the torque is most important later in the downswing (where the velocity is the highest), we will pick the dip just before impact. That will be our reference velocity. For this study, we will start by finding a constant torque that, when reduced by the torquevelocity curve and applied to SwingPerfect, gives a 100mph clubhead speed. (Due to the shape of the curve and the "granularity" of setting torques in SwingPerfect, we will slope the torque reduction for the first two segments of the swing.) This takes some iteration, because changing the torque gives a different reference velocity, which gives a different torque reduction. Then we will take that constant torque, apply it to SwingPerfect with different head weights, and again iterate on torquevelocity to get a consistent clubhead speed. Here is the graph we get when we go through the computation of carry distance. The green curve is the Daish study, and the red curve is the original computer simulation. The blue curve is the modified simulation, where we reduce torque based on the torquevelocity curve. Accounting for the biomechanical reduction of torque does not give us the Daish (nor Cochran & Stobbs) results, but it does cut the headweight advantage in half. 

I
tried a variation of the torquevelocity study, where I used the entire
reference velocity to reduce the torque during the entire downswing.
This is not completely valid, because it takes 100150 milliseconds for
the angular velocity to get up near the reference velocity. For that
first 100150 milliseconds, the torquevelocity curve does not provide
nearly as much reduction. The simulation above accounted for this, but
let's see what happens if we apply the full reduction for the full
downswing. What happens is shown in the graph. The carry distance falls off as the clubhead weight is reduced. So the torquevelocity curve is capable of reducing the clubhead speed down to the Daish level, or even lower. But we have to apply more of it than is physically accurate. But wait! This suggests something else that might be going on. In fact, it is very likely going on. What this study does is tone down the torque applied to body rotation early in the downswing. That may well be happening, even if the torquevelocity curve is not the reason. It is probably unreasonable to expect full body torque to be applied the very first millisecond after the transition from backswing to downswing. Mackenzie's biomechanical model has builtin a time constant that says it takes muscles 60msec to build up close to full strength. If considerations of balance in the transition slows this down more, the torso/shoulder torque might take 100150msec to build up  and the result might well look like Daish's study. 

ConclusionsHere are the conclusions I draw:

