Newton and the Divot:


Dave Tutelman  --  January 28, 2014

This article stirred up much more comment than most I write. I got a lot of email with very interesting questions and comments. And Fredrik Tuxen of Trackman told me that I had misinterpreted Trackman's definition of Attack Angle as stated on their web site. Actually, I used it completely literally. The definition needs a little work to state exactly what Trackman measures and reports.

So it seems appropriate to add a page of follow-up and correspondence, because a lot of the questions and comments require more than just a small tweak on the existing article. (I did that where I could, but it left over a lot worth discussing.) So here it is.

Trackman's definition of Attack Angle

The most controversy, and the biggest changes needed to my article, dealt with the way Trackman defines what they report as Angle of Attack. At the time I original wrote this article, the definition on the site was:
Attack Angle - The vertical direction of the club head’s center of gravity movement at maximum compression of the golf ball

Today in 2018, it has been slightly updated to:
The up or down movement of the club head at the time of maximum compression. Attack angle is measured relative to the horizon.
It is a different wording, but still says the same thing.

Fredrik Tuxen, the CTO of TrackMan, responded to the article in email to me and a post to the Facebook group for Professional Golf Instructors. The latter post said:
TrackMan's angle of attack is taken at point of time of max compression of the ball. However, ONLY by using pre-impact data. Meaning the attack angle from TrackMan is very close to first contact with the ball but shifted app. 0.3 deg due to the 1/4000 second time difference between first contact and maximum compression of the ball.

Tutelman has unfortunately misunderstood the definition of TrackMan's angle of attack, but he is excused it was not sharp enough.
TrackMan will update the definitions of attack angle and club path to avoid future misunderstandings.

Tutelman's 'Newton divot' is an insightful description in the consequences of conservation of linear momentum for the oblique impact between club and ball.
He followed up with a video Skype session with me, covering the details of how they measure and report Angle of Attack. Here is my understanding of it, explained in the context of a diagram. Fredrik says this is a correct interpretation.

TrackMan picks up the clubhead well before impact, and tracks it to the ball, getting as close as possible to the ball while maintaining accuracy. In the diagram, it tracks through positions 1, 2, and 3. (Actually, it notes a lot more than three positions in this interval.) Those positions follow an arc, the red dotted line, which is mathematically fitted to the clubhead's measured positions as it approaches the ball.

TrackMan is capable of measuring the time and position of  the beginning of impact very accurately. It extends the clubhead along that same arc (the red dotted line), to where it would be without any clubhead deflection in the middle of impact at the point of maximum ball compression, shown as position 4 in the diagram. The slope of the arc at position 4 is the Attack Angle reported to the user. Note that:
  • While the clubhead is indeed deflected downward during impact, that deflection is not considered part of TrackMan's AoA measurement.
  • Since the clubhead had been traveling a curved arc, the reported AoA is a bit more upward than the pre-impact AoA. TrackMan has found this difference to be 0.3-0.4 degrees.
I went back and recalculated the tables. The calculations are much easier using this definition. I was able to dispose of half the green math section of the article. The results are now on the first page of the article.

Thanks, Fredrik.

While the short definition of Attack Angle on the TrackMan web site has not changed appreciably, those wanting to dig deeper can find the correct details in their new article on "Club Data Definitions". This article acknowledges momentum transfer's significant contribution to clubhead movement during impact, and explicitly excludes the influence of the ball from the reported measurement.


There were quite a few emails and forum posts with worthwhile questions and comments. Let me address them.

In what follows, I don't identify the commenter. I present the comment that was made, and respond to it without attribution. If you recognize your own comment and want to be credited, let me know and I'll be glad to do it.

I have not reread your article but my first run through doesn't leave me with any more information about why you wrote it and what it is suppose to accomplish. Nice charts and physics discussion but isn't what you presented taking the phrase "angle of attack" and nit picking it?

I enjoy discussions like you presented but I cannot take that to the golf course as the guys I play with would look at me and proceed to throw me into one of nearest ponds.
I write articles for different audiences, and very few articles are aimed at all of them. From your comment, I suspect you are a golfer with a technical interest, but not a clubfitter, club or component manufacturer, instrument designer, professional golf instructor, or researcher. This article is probably of marginal interest to you, but I've gotten a lot of very interesting discussions from instructors and clubfitters, and instrument designers.

When I go out on the course with my own friends, there are precious few that I could mention this to without the chance of a dunking, so I understand what you say.

Even so, you might take this away from your read: If you hit a ball from the ground with any club at all, you better leave at least a scuff mark on the ground after the ball -- and a real divot with a middle to short iron. If your club never touches the ground, that doesn't mean you picked it clean; far more likely, you hit it thin.

Your interpretation of AoA is measured just before impact. So what is that "just before"? We need two points so we need a time frame, but how large a time frame? It would seem to me that the value here is dependent on the sample rate or fps of the camera used!
Absolutely right! In fact, even more points at a high frame rate give us a more accurate estimate. And that is what TrackMan does. Their sampling rate (20,000 samples per second) is comparable to the fastest cameras. They use this to fit an arc to the clubhead path. If you factor in the ability to estimate the time of impact very closely, you can extrapolate that arc to either the moment of impact or the moment of maximum compression. They choose the latter.

But even more important, does it really matter to the ball and the ball flight? I would say it doesn't. In fact it's the real movement of the head during the impact period (till max compression) that is important.
Depends what you are trying to do with the data. In my case, I was doing momentum transfer calculations. Momentum must be computed before impact begins and after impact ends. Anything that happens during impact is just going to confuse the calculations. Let's take a closer look at what goes into a momentum transfer.

I had always assumed that the duration of impact was so short that there wasn't much change in clubhead presentation during the impact interval but that appears not to be the case.
Definitely not the case! Momentum transfer is a wonderful thing. 1/2000 of a second is not very long, but 2000 pounds is a lot of force. To get a feel for why this matters, we need a concept called "impulse", which I haven't mentioned yet.

Let's look at how momentum comes about. One of Newton's three famous laws is F=ma. We should examine that a little more closely. Acceleration is the change of velocity over time. So we can rewrite F=ma as
F  =  m


Where the delta represents a change or interval. We rewrite this as:

F Δt  =  m Δv

We recognize the right side as the change in momentum. What the whole equation says is, "The change in momentum is equal to the force acting on a mass times the time interval the force is exerted." The left side, the force times the time interval, is called "impulse", and we see that momentum change of a free body is equal to the impulse on that body. So impulse = change in momentum. (For a more complete explanation, try this.)

Now think about momentum in terms of impulse. Two thousand pounds of force applied for 1/2000 sec gives the same momentum transfer as if it were one pound for one second. Much easier to think about -- and not negligible at all.

The pros hit the center of the club face, and the tables in the article are based on the pros. Would you see less effect for the average golfer?
That's a great question! The effect is based on momentum transfer, and the pros transfer momentum very efficiently. So perhaps they will see more of a deflection than you or I would. Let's look at some of the factors involved in the momentum transfer:
  • Momentum is MV, mass times velocity. Everybody plays with the same ball mass and the clubhead mass doesn't have a lot of variation. But the pros generate more clubhead speed and especially more ball speed. (Their center strikes have a better smash factor.) So that suggests more deflection for the pros.
  • The downward deflection of the clubhead is based on upward motion of the ball. The pros hit the ball with a flat or bowed wrist, resulting in a fairly low trajectory. I see a lot of somewhat cupped wrists among my playing companions, which will result in a higher launch angle. This means a proportionally larger vertical component of momentum, which will give the duffers more downward deflection.
  • True, the average golfer is not a center-face striking machine. But it's worse than that. Often enough, they won't get it on the face at all -- something you almost never see from the pros. A thin shot that is at least partially off face (partially bladed or worse) will not generate much downward force on the clubhead. And you can see it as a momentum transfer thing by watching the worm-burner scoot along. You hardly ever see that from a pro.
So I see three factors: one where the pro would experience more deflection, another where the average golfer might see more deflection, and a third where an average golfer would sometimes encounter a situation of almost no deflection.

Does Flightscope have the same problem? What is its definition of Angle of Attack?
Henri Johnson, CEO of Flightscope, told me through a mutual friend that their definition is:
"The angle between the velocity vector and level ground, just prior to impact, is AOA."

That is the definition that I understand, and the one I use here. I hope it is in fact the one they use in their measurement.

The reason for my uncertainty is, you won't find the definition if you check the Flightscope web site. Instead, you will find a video on the subject. The definition implied in the video has a final sample just before separation (not just before impact), which would include the downward deflection of the clubhead. That is not in accordance with their definition.

Concerning the TM tour average numbers, are you aware that TrackMan reports AoA and clubpath from the geometric center of the clubhead and not the clubface? I have a sense that this could change some of the calculations you have preformed.
The truly relevant position on the clubhead would be the center of gravity (GC), not the geometric center and not the clubface. So neither TrackMan nor I get it exactly right. But I think we both do pretty well. Here's why:

The center of gravity is usually pretty close to the geometric center of the clubhead. Not exactly the same, but close. So TrackMan will not be far off.

I did pick my points near the face, so your comment is on target. But:
  • For the irons, the CG is not far from the face at all. Any error due to picking a point on the face is quite small.
  • For drivers the problem is potentially much larger, because the CG is well back from the face. (I mentioned it, in fact, as the last bullet point in the section "Even the driver?") For that reason, my case study for a driver involved center impact, where the clubhead did not rotate. Any up/down motion of the face in that particular case corresponds to an up/down motion of the CG.
So you raised a valid concern. I had the same concern, and believe I have it covered.

Momentum is not conserved in the club/ball collision. Some energy is turned to heat or sound, while compressing the ball and clubface. You should include COR in your calculations, or they will be called into question. 
You are confusing conservation of momentum with conservation of energy. (I ran across the same mistake last week in a book on golf physics by John Zumerchik, so you are not alone.)

When you set up the equations for a collision, you need both conservation of momentum and conservation of energy. Both! The conservation of energy equation involves COR and energy losses. Momentum, however, is conserved without loss.

In the calculations here, only momentum needs to be considered. We are given the clubhead velocity before impact and the ball velocity after impact. Therefore, the COR has already been accounted for; the ball's velocity reflects the energy losses. The problem to be solved here is just conservation of momentum.

While the objection has been answered, let's assure ourselves that we can indeed conserve momentum perfectly and still reflect energy loss. Here's an example. Consider two possible two-ball collisions, one lossless and one very lossy.

Lossless collision
COR = 1.0
Balls bounce perfectly
Lossy collision
COR = 0
Balls stick together

Before Collision
After Collision
Before Collision After Collision
2M׽V = MV
MV2 MV2 MV2 2MV2/4 = MV2

The important thing to notice is: in the lossless case, both momentum and energy are conserved. That is, they are the same both before and after. But in the very lossy case (COR=0), momentum is still conserved but half the kinetic energy in the system is lost. So it is OK to ignore losses if the problem is simply conservation of momentum. Losses must be accounted for when you do conservation of energy.

Given that the clubhead slows down during impact and the shaft is bent forward just prior, I was wondering how much of the lowering (and delofting) may be due to just unbending of the shaft. Or do you think that in this short period of time the clubhead really acts more like a free body disconnected from the shaft?
The latter, definitely! I address this in other articles, and in great detail in my article on gear effect. The influence of the shaft on the clubhead during impact is almost nothing, relative to the other forces acting on the clubhead. Consider: it takes an inch of deflection of the shaft to produce a two-pound spring force, but the force between ball and clubhead is of the order of two thousand pounds. So the inertial forces are three orders of magnitude greater than the forces the shaft is exerting.

In one iron shot in particular, the ball was hit towards the toe. The face opens during and after momentum transfer, which causes a motion that has a circular pattern for the reference point on the club. The reference point on the club hence is moving downward, causing the apparent perception of a "downward movement".
If you mean the measurements from the iron in the animated GIF, then you are correct. There is in fact plenty of actual downward movement, it is not limited to just apparent movement. But the motion of the toe -- a little downward and and even more "backward" motion -- exaggerates the angle of deflection. Probably for that reason, the deflection angle measured from the video frames is 12, about twice the value calculated for Tour strikes. But make no mistake, there is plenty of real downward movement in the video -- just not as much as it looks like.

It is worth saying more about this. In order to take accurate measurements from frames of a video, you can't just use random videos. Mea culpa! I used random videos, even knowing the problems. I was looking to demonstrate an effect, which I would quantify by calculation. I don't have a high-speed video camera, and had to make do with what I could find on the Internet. If you looked closely, you noticed the measurements from the videos were inconsistent with the calculated values. The iron video measured at twice the calculated value (12 vs 6), and the driver video measured at half the calculated value (3.2 vs 6).

In order to get an accurate measurement, here are a few things you should be careful about before you even start to film:
  • The camera should be pointed perpendicular to the target line.
  • The camera should be far enough from the clubhead so perspective distortion is minimized. (That is, motion closer to the camera should measure almost the same as motion farther from the camera. If the camera is within a few feet, closer will measure substantially more than farther.)
  • The clubhead should be marked with camera-visible dots, preferably dots with sharp corners, so clubhead position can be determined to within a pixel.
  • The position of the impact on the clubface should be controlled -- or enough clips taken so the right position can be selected -- to avoid the valid criticism applied to my measurement.

One point can not be used to measure the 3-dimensional movement of the clubhead. If you take a club and rotate it around its CG, then that point would be moving but the CG would not! Two points would be better, as you could use the distance between them as an indication of movement towards the camera. To get the correct 6DOF data from any object in one video a minimum of 3 points is needed and those point have to be arranged in a correct way.
All very true, but not a big issue here. 6DOF (jargon for "six degrees of freedom") includes three degrees of rotation and three of linear motion. I maintain we only need two degrees of linear motion for this exercise, so a single point is sufficient. Looking at my the previous response:
  • Rotation of the clubhead at impact does not need to really be measured, just avoided in the frame we use. That will depend on the position of impact on the clubface -- as noted above. We need to pick neutral videos -- near-center impact -- to do our measurements.
  • Multiple dots serve two purposes: rotational degrees of freedom (which for our purposes only need to be detected, not measured; multiple dots not needed) and motion toward or away from the camera.
  • If the camera is perpendicular to the target line, then motion toward or away from the camera is not an issue. And the camera has the other two dimensions covered.

Last modified - Aug 8, 2018