Hi Dave,
I have no idea if the chs was "at impact" or "maximum" nor does the competitor,
it was done on the Titleist launch monitor in Carlsbad Ca.
Maybe someone out there is familiar with their set up or can check.
I will try to find out.
This long drive competitor is certainly "skilled" in fact you might even call him the "Jack Nicklaus" of long drive guys.
What I am after of course is the magic formula.
If we increase club head speed due to a lack of mass but we increase ball speed due to a presence of mass then what is the formula where we achieve a peak?
And how is this affected by length, and or acceleration or I guess velocity.
Of course, the analytical "holy grail".
Let's see if I can beat TFlan and others to the punch in pointing out that reality -- and especially the reality of the average duffer's swing -- may introduce much bigger variations than physics. "The difference between theory and practice is much bigger in practice than in theory." :-)
Anyway, the points of this post:
(1) The basic formula I presented is as good a first approxmation as we have. It really does work!
(2) It misses explaining your data BIG TIME! So we need to look at what could be different.
OK, here's the detail on those points:
(1) THE FORMULA DOES WORK
Hey, it's really basic physics, stuff that hasn't changed in centuries.
And it works for the golf swing as well. An anecdote to support that:
In 1997, when Tiger was in his first full season on Tour, there was a discussion on rec.sport.golf about his ball speed. I posted that it had to be around 178mph. I got a private email from someone in Titleist R&D, saying basically, "How the hell did you know that? We measured it at 177mph, but never published anything about it." I wrote back that I figured it from Cochran's formula and some pretty easy guesses at what went into the parameters. I got back a note to the effect of, "Hmmm. Very good."
So it certainly works for a really good swing.
(2) ...BUT IT DOESN'T EXPLAIN THE RESULTS DAVID POSTED
For the four drivers tested, the ball speed falls well short of what the formula predicts. It is 2.5% short for one driver (probably at least 10 yards worth of distance). For the other three, it is about 10% short.
Let's look at all the things folks have suggested so far -- and maybe a new hypothesis or two -- and try to quantify how much they might contribute to this 10% error.
ACCELERATION AT IMPACT - We've been beating this to death for a couple of days now. Since the collision isn't instantaneous, but lasts a half millisecond, acceleration can have some effect. But the effect is less than 0.2%. Not a factor here!
DECELERATION BEFORE IMPACT - If the measured clubhead speed is a maximum that occurs before impact, that could have an almost unlimited effect on the results. But it's very hard to believe that this guy (David calls him 'the "Jack Nicklaus" of long drive guys') gets to an early maximum that is 10% faster than his impact speed.
Look at the graphs Bernie referred to, the ones comparing Bobby Jones' acceleration and velocity curves to those of an "ordinary golfer". The ordinary golfer has a few swings that look like this could explain it. But not Bobby Jones. It was hard to find a velocity maximum higher than his impact speed. True, he had almost no acceleration at impact, but he didn't have any significant deceleration either.
MASS OF THE GOLFER - The clubhead isn't floating in air when it impacts the ball; it is supported by a shaft, and a golfer at the other end of the shaft. Do we have to include some or all of the golfer's mass along with the clubhead?
It is generally believed that "the shaft is a string at impact"; that is, during the very short period of impact, nothing can be done at the grip end of the club to affect the clubhead. But even if you don't accept this, consider: any mass that we add to the clubhead to account for this will increase what the theory says the ball speed should be. And we already have a theoretical ball speed that is higher than what was observed. So this does not explain the discrepancy; in fact, it makes the discrepancy worse.
So, if you believe the common "string at impact" wisdom, this is not a factor. And if you don't believe it, we have more to explain than we did before.
LOFT, LAUNCH ANGLE - Several people have pointed out that we really have a 2-dimensional impact here because of loft. The formula is based on a one-dimensional collision (zero loft). True enough. How much error would this contribute?
The launch angle extremes were 11.5* and 12.5*. Let's assume (probably contrary to fact) that this whole angle can be attributed to the effective loft at impact, and none of it to angle of attack. If so, that might contribute as much as 2% to the error.
But that assumption is not likely to be true. A good long driver wants to lower the spin rate somewhat, so the launch angle comes from a lower loft and higher angle of attack. If we assume the loft at impact is more like 8*, then it contributes only 1% to the formula error. This is somewhat significant, but not a major fraction of even the smallest error we observed. So it might conceivably be part of the story, but only a small part.
OFF-CENTER IMPACT - This could give big losses of ball speed. But that isn't likely to be the problem for this guy. Of course, it would be nice if impact tape were used to verify that it isn't the problem.
LOW COR - There might be reasons that the COR is lower than tested on the USGA pendulum device. Let's look at a few:
* Clubhead speed: COR is sometimes lower at higher clubhead speeds. (I've never seen a study that reports it higher, but I have seen studies that have it a bit lower.) With the extreme clubhead speed of a world class long-driver, this might become a serious issue.
* Ball compression: One reason for lower COR at higher clubhead speeds is that the energy losses due to ball compression are not linear. You lose relatively more as you compress the ball more, creating higher than proportionally-more losses. This may or may not be the same effect as the first bullet.
* Ball-clubface matching: I don't know how much frequency matching will contribute. My intuition tells me it won't get to 10% -- but we know how good intuition is when it comes to golf physics.
Why don't I think it will be 10%? Because the resonance of the ball at its natural frequency is quite damped. That means that the response is "broad"; it isn't that much lower as you move away from the natural frequency. But I don't know either the frequency or the damping factor of the ball, so I'm just guessing here.
There are probably other factors as well. I hope there are -- and not just measuring instrument error -- because I don't think these factors alone will get us to the 10% we need to explain the discrepancy.
Cheers! DaveT
