Newton and the Divot

Dave Tutelman  --  January 19, 2014
Not many people seem to be aware that the clubhead is deflected downward at impact. The physics is pretty easy, but the fact is obscure. It is a partial explanation of why your clubs have to take a divot. Beyond that, it may or may not be useful, but is certainly interesting.

Let's start this discussion in the same context I first became aware of the effect.

How many times have I listened to Peter Kostis narrate CBS TV's Konica-Minolta SwingVision slow-motion videos like the one at the left? (Kostis narrating a Tiger Woods shot, a short clip from a video on YouTube.) Almost every time, the commentator will say something about "hitting down through the ball" and "taking a good divot". It is an article of faith that a divot taken after the ball is necessarily an indicator of a downward strike.

At first glance, this would seem to be not just an article of faith, but an article of geometry as well. How can you take a divot at all if the clubhead is not traveling notably downward coming into impact? You can't, right?

Look again at the video. Yes, the clubhead is descending as it approaches impact. But at a rather shallow angle. Notice the difference between coming into impact and as the ball is leaving the clubface. In particular, notice that the clubhead is not only traveling slower (a well-known effect), but also on a more downward angle!

The more you study videos of impact, the more you realize that this is the norm, not a peculiarity of the Kostis-Woods example. For instance, check out the video on the right of Luke Donald's iron shot. (This is NBC, not CBS, and the trademark name of their slow-motion is "NBC HyperMo".) Here the ball is teed up, and the angle of attack is so shallow coming into impact that it is easy to believe there will be no divot. But the clubhead is plainly deflected downward at impact, and a shallow divot is the result.

It didn't take me too much thought to understand why this should be, once I realized that it is a very usual observation. In fact, the principle is easily understood from high school physics, and the calculations are no more difficult than most homework for Freshman Physics 101. Let's take a look.

Why it happens

(The illustrations in this section are created from frame snapshots of an excellent video of slow-motion impacts made by the Biomechanics Department at Manchester Metropolitan University.)
The key to this puzzle comes from two very basic principles of physics, both directly from good ol' Isaac Newton.
  • F = ma
  • Every action has an equal and opposite reaction.
The picture to the right shows a golf ball at maximum compression on a clubface. The ball is about to take off in the direction of the green vector. It is doing so under the influence of a force. The net force during impact has to be in the direction of the launch angle, because of F=ma. The netting out of acclerations during impact gives us a velocity in the same direction as the netting out of forces. So the green vector F is the net force the clubhead exerts on the ball during impact, as well as an indicator of the launch angle.

But ol' Isaac also told us that every action has an equal and opposite reaction. The fact that the clubhead exerts force F on the ball, means that the ball exerts an exactly equal and opposite force on the clubhead. That is the red vector R.

It is common practice in physics to break down vectors into their vertical and horizontal components. We have done that in the next picture. F and R have been "resolved" (broken down) into:
  • A horizontal force accelerating the ball downrange, Fhor
  • A vertical force accelerating the ball upward, Fvert
  • A horizontal force slowing the clubhead, Rhor
  • A horizontal force deflecting the clubhead downward, Rvert
The last one, Rvert, is the reason the clubhead turns downward at impact. Any upward acceleration of the ball -- any launch angle greater than zero -- causes a downward acceleration of the clubhead!

How much happens

OK, we can see there must be a force deflecting the clubhead downward. But the clubhead is heavy, and there is a golfer on the other end of the shaft. So how big could the deflection be? We could calculate these effects -- and we will later. But first...

Let's do some measurement of the high-speed videos to see if this phenomenon is large enough to matter. In particular, let's see if it would cause a divot to be taken when there would otherwise be none.



I made an animated GIF from several frames of the video from Manchester Metropolitan University. It traces a point on the clubhead from before impact to after the ball is on its way. The yellow trace shows the movement of the clubhead. It is decidedly deflected downward during impact. The angles of attack during various intervals are:

Clubhead approaching ball
-0.4
Average angle during impact
-6.3
Average angle from first contact to maximum compression
-4.3
Average angle from maximum compression to separation
-8.4
Immediately after separation
-12.5

Points to note:
  • The angle of attack approaching impact is -0.4. That is hardly "hitting down on the ball"; it is a nearly horizontal "pick".
  • By the time the clubhead leaves the ball, the downward angle is more than 12. That is pretty steep for an angle of attack, and would definitely leave a prominent divot. Given that the original angle of attack is nearly zero, this can truly be called a "Newton's Divot".
  • At maximum compression, the downward angle has changed by almost exactly half the entire change during impact. So half the change occurs before maximum compression, and the other half after.[1] (This doesn't seem important now, but we will use it later when we examine the significance of TrackMan measurements of PGA Tour data.)

Even the driver?

Most physics analyses of driver impact assume everything happens in a horizontal direction. With the lower loft of drivers, this is not a bad assumption -- if you are looking to find only ball speed. But even a drive has a non-zero launch angle, so there is some downward deflection, at least in theory.
Is there enough to see in a video?

Here is a video that shows it. Taking frame snapshots and measuring the angles, we get:

Clubhead approaching ball
-1.8
Immediately after separation
-5.0
Launch angle of ball
10.3

To get a feel for the angles, we should "normalize" them to the angle at which the clubhead approaches the ball. When we do that, we get:

Clubhead approaching ball
-1.8 - (-1.8) =
0.0
Immediately after separation
-5.0 - (-1.8) = -3.2
Launch angle of ball
10.3 - (-1.8) = 12.1

This has a much better "feel of reality" to it. We will compute values like this later.

Unlike videos of irons, which almost always show a downward deflection, driver videos sometimes do and sometimes don't. Why should this be? Here are a few possibilities:
  • The best drives are struck with an upward angle of attack, so the deflection may merely level out the head motion from its original upward direction.
  • If the ball is hit above or below center, the clubhead rotates face up or face down. If the strike is above center -- which gives the best drives, so pros try for this -- then the face is moving upward after impact, with respect to the center of the clubhead. So the center may be deflected down, but the face may be moving up relative to the CG and making it appear that the clubhead is not deflected. The irons do not have nearly as much offset between center of gravity and face, so clubhead rotation does not provide much vertical clubface motion.

TrackMan Tour averages

The original posting of this article (1/22/2014) was based on the definition of Attack Angle published on the TrackMan web site. It turns out that the definition is not an accurate description of how TrackMan measures Attack Angle. We now have a more accurate definition, and I have recomputed the tables based on it.
On January 11, 2014 Adam Kolloff pointed out to me that in 2013 the average driver angle of attack on the PGA Tour was negative. That struck me as odd. Physics says the drive will be longer (higher launch for a given spin) with a positive angle of attack. Therefore, I would expect the best golfers in the world to be trying for an upward angle of attack.
PGA TOUR AVERAGES
(from  WWW.TRACKMANGOLF.COM)


Club Speed
(mph)
Attack Angle
(deg)
Ball Speed
(mph)
Launch Ang.
(deg)
Driver
113 -1.3 167 10.9
3-wood
107 -2.9
158 9.2
5-wood
103 -3.3 152 9.4
Hybrid 15-18
100 -3.5 146 10.2
3 Iron
98 -3.1 142 10.4
4 Iron
96 -3.4 137 11.0
5 Iron
94 -3.7 132 12.1
6 Iron
92 -4.1 127 14.1
7 Iron
90 -4.3 120 16.3
8 Iron
87 -4.5 115 18.1
9 Iron
85 -4.7 109 20.4
PW
83 -5.0 102 24.2
Adam sent me a copy of his source information: a table from the TrackMan web site. Selected columns are reproduced here as a web table.

Sure enough, the average Angle of Attack (AoA) for the driver was -1.3 in 2013. For a while, I considered downward deflection as an explanation. This speculation was reinforced by another bit of information Adam sent, TrackMan's definition of Angle of Attack:
Attack Angle - The vertical direction of the club head’s center of gravity movement at maximum compression of the golf ball

It turns out that this is not quite the operative definition that TrackMan measures. It is more like an extension of the clubhead path into the middle of impact, without the clubhead deflection. The exact method is given on the next page.

Given the data in the four columns of the Trackman chart, it is not a difficult physics problem to calculate the downward deflection angle of the clubhead.
Accounting for Downward Deflection

TrackMan
Attack
Angle
(deg)
Pre-Impact
Attack
Angle
(deg)
Impact
Deflection
(deg)
Divot
Potential
(deg)
Driver
-1.3 -1.6 -6.6  -7.9 
3-wood
-2.9
-3.2 -6.1 -9.0
5-wood
-3.3 -3.6 -6.0 -9.3
Hybrid 15-18
-3.5 -3.8 -5.8 -9.3
3 Iron
-3.1 -3.4 -5.4 -8.5
4 Iron
-3.4 -3.7 -5.4 -8.8
5 Iron
-3.7 -4.0 -5.6 -9.3
6 Iron
-4.1 -4.4 -6.0 -10.1
7 Iron
-4.3 -4.6 -6.2 -10.5
8 Iron
-4.5 -4.8 -6.4 -10.9
9 Iron
-4.7 -5.0 -6.6 -11.3
PW
-5.0 -5.3 -6.9 -11.9
The results are in the next table. (The math itself is done in a later chapter; you don't have to follow the math to understand the results.) Here is an explanation of each of the columns, and how to interpret them.
  • Trackman Attack Angle - The AoA as defined and recorded by TrackMan. This column is just transcribed from the previous table.
  • Pre-Impact Attack Angle - The calculated AoA as we normally think of it, the angle just before contact with the ball. (Working backwards from the TrackMan definition, it is the TrackMan Attack Angle minus a slight path curvature of about 0.3 [2])
  • Impact Deflection - The calculated downward deflection of the clubhead due to impact, as an angle in degrees.
  • Divot potential - I thought it would be amusing to include the vertical angle of the clubhead after impact. This, not the AoA, is the number that really determines the potential for cutting a divot from the sod.

Discussion

  • I still find it odd that the AoA for the driver remains negative. But that is what the data says! Perhaps the average Tour player (remember, these are average numbers, not just the big hitters) are going for solid contact at the expense of distance. The debate continues between the "bomb and gouge" strategy and the "fairways and greens" strategy. This suggests the latter is adopted by the average Tour player.
  • Suppose we look at competitions where the players need more distance. Here are a couple.
    • The LPGA Tour players do not have the clubhead speed that the men do. If they are going to get distance from their driver, they need distance mechanics in their swing. TrackMan has published their averages as well, and the driver AoA is +3.0. That's a very upward swing path. And the upward AoA is only for the driver; they treat the rest of the clubs as control clubs and have a negative AoA.
    • Distance is everything in the long drive! You get six tries. Only one has to hit a very wide "fairway", but that one better be lo-o-o-o-ong! Fredrik Tuxen tells me that TrackMan measured everything at the recent ReMax World Championship. Those guys have around a +5 AoA, and at least one close to +10. Now that is high launch angle with low spin!
  • The downward deflection of the clubhead is within a degree of -6 for every club. I suppose it makes some sense. As loft increases, the launch angle increases, so the downwards component of the reaction force increases -- at least as a fraction of the total reaction force. That would make one expect the deflection to increase with loft. However, while the downward component's share is increasing, there are at least two things happening that are limiting the downward deflection:
    • The ball speed is decreasing. So, while the vertical share of momentum may be greater, it is a piece of a smaller total momentum.
    • The clubhead mass is increasing. The 200g driver head becomes 291g by the time it gets to the PW. Since F=ma, an increase in mass requires a decrease in acceleration. And less downward acceleration means less deflection.
    The result: The downward deflection increases for each club in the irons by a very small amount (from 5.4 in the 3-iron to 6.9 in the PW). It is a steady, monotonic increase averaging about 1/5 of a degree per club.
  • The Divot Potential column also makes sense -- mostly. As the loft increases, the downward angle of the clubhead after impact also increases. That comes from an increasing attack angle and a substantial downward deflection. And that downward deflection, at least in the irons, increases slightly as the loft increases.
  • The one anomaly I see in the Divot Potential column is that the driver is moving downward after impact at an angle of almost 8. How can it do this and not take a divot? I can't cite a real study, neither experimental nor analytical, but let me venture some fairly good guesses.
    • The ball sits up on a tee, not on the ground. So the clubhead has some room to move downward without hitting the ground, as long as the downward motion does not continue too long.
    • The pros can control where they hit the ball on the clubface, and the best place to hit a driver is high on the face. This gives a higher launch angle and lower spin than nominal for the driver, resulting in more distance. That means that the clubhead is turning face-up after impact; moreover, because the CG is so far behind the face, the face is moving up as well as turning up. If the clubhead is going to strike the ground at all, it will be the tail of the clubhead -- which is not going to dig but just bounce. For a rather graphic example, see Figure 4-7 in my article on Gear Effect.
  • The other side of the Divot Potential coin is how far downward the clubhead continues. Looking at high speed video, the downward deflection is fairly short term, then the downward motion becomes limited by shaft rigidity and bounce from the ground. Consider:
    • The shaft is almost a string during impact; I have explained that in several places in my writings. However, as milliseconds build up after impact, the influence of the shaft -- and a large mass of golfer at the other end of the shaft -- will limit the downward motion of the clubhead.
    • If the ball is sitting on the ground, then resistance from the ground may limit downward motion even before the shaft does. A back-of-the-envelope calculation shows the clubhead moving downward at about a quarter inch per millisecond. It reaches the ground pretty quickly at that rate, and ground resistance will lessen the downward motion.
    So the clubhead motion in the 20 milliseconds after impact is pretty complex, and I'm not going to try to analyze it more than I just have -- at least not for now. But I will note...
  • Bobby Clampett, in his book "The Impact Zone" (St. Martin's Press, 2007), hammers at the point that the swing bottom (the lowest point of the clubhead's travel) should occur well after striking the ball. He maintains that scratch and tour players have a swing bottom four inches in front of the ball. In fact, he cites an informal study showing that each inch in front of the ball is worth about four strokes in the handicap. The downward deflection probably contributes in some way. Think about it. With that much down angle just caused by ball-on-face impact, you need a diet of very thin or somewhat fat hits to avoid a forward swing bottom. If you catch the clubface solidly, the reaction force will guarantee the clubhead is traveling downward when it leaves the ball -- even if you don't have an aggressively downward AoA.

The bottom line

The clubhead is definitely deflected downward at impact. The divot taken by an iron is at least as much due to the downward deflection as it is to the angle of attack before the clubface contacts the ball. In fact, PGA Tour players have rather shallow angles of attack, but their clubheads are traveling much more downward after the ball is gone. There are other things that are explained by downward deflection, such as the negative angle of attack reported by Trackman for driver strikes; that seems to be an artifact of Trackman's definition of AoA and the fact of downward deflection of the clubhead.

Calculating the downward deflection

We can calculate the downward deflection if we are given the clubhead speed and angle of attack before impact, and the ball speed and launch angle after impact.

If you are a mathophobe, you don't have to read this section. The results can be understood without wading through the math. But if you want to investigate this sort of thing yourself, you should really understand it thoroughly. From a math and physics standpoint, it is quite easy. The physics is freshman physics 101, and the math is high school algebra and trigonometry.

Velocities and angles:
Here are all the velocities before and after impact.


U
a velocity before impact
  
V
a velocity after separation
c
a subscript designating the clubhead

b
a subscript designating the ball
x
a subscript designating horizontal rightward motion
(leftward is negative)

y
a subscript designating vertical upward motion
(downward is negative)

Other notation we will use:
P
Total system momentum. Since momentum is a vector quantity, it has separate horizontal and vertical components which take subscripts x and y.
M
Mass. Can take c subscript for clubhead, or b subscript for ball.
 - Clubhead mass for a given club (say, a 5-iron) does not vary much among most manufacturers. For each club, I took a consensus number from several catalogs. (E.g.- 200g driver, 256g 5-iron, 291g PW.)
 - All competitive golf balls are right at 46 grams.
L
Launch angle.
D
Deflection angle.
A
Pre-impact angle of attack

We know some of these -- e.g., from launch monitor or Trackman readings -- and will calculate the rest of them from the principle of conservation of momentum. Momentum is the sum of the mass-times-velocity for every mass in the system. Conservation of momentum says that the total momentum of the entire system -- all the masses involved -- is the same after impact as it was before impact.

Find system momentum before impact:
Momentum is just mass times velocity. The only thing moving before impact is the clubhead. So the momentum's horizontal and vertical components are those of the clubhead:

(Equations 1)Px  =  Mc Ucx  =  Mc Uc cos A
Py  =  Mc Ucy  =  Mc Uc sin A

Equate the momentum to post-impact momentum:
Impact happens. We get impact and separation. Now we have to find the post-impact velocities that satisfy conservation of momentum. That means that the Px and Py we calculated above must be the same Px and Py after impact.

Since momentum is mass times velocity:

(Equations 2)Px  =  Mc Vcx + Mb Vbx  =  Mc Vcx + Mb Vb cos L
Py  =  Mc Vcy + Mb Vby  =  Mc Vcy + Mb Vb sin L

We already know Px and Py; conservation of momentum says they must be the same as the momentums we calculated before. In fact, the only things we don't know here are the head velocities Vcx and Vcy. For each of these unknown velocities, we have an equation that  is easily solvable for the head velocity. The solutions are:

(Equations 3)Vcx  =  Px / Mc   -   (Mb / Mc) Vb cos L
Vcy  =  Py / Mc   -   (Mb / Mc) Vb sin L

The velocities give us the clubhead path and deflection angle:
Now we have the clubhead velocity, both magnitude and direction, after impact. The direction is simply arctan(Vcy/Vcx). If you don't see this immediately, look at the the "Just after separation" picture above.

The deflection of the clubhead's path is, by definition, the difference between the direction after impact and the direction before impact. Expressed mathematically:

(Equation 4)D  =  arctan (Vcy/Vcx) - A

Acknowledgements

I would like to thank Adam Kolloff for getting me off my butt to write about this. He showed enough interest to come up with some data that requires knowing about downward deflection in order to explain it.

Also, thanks are due to Fredrik Tuxen, founder and CTO of Trackman, for spending time with me to explain how they measure Attack Angle.


Notes:

  1. The measured numbers are 3.9 of deflection occurring before max compression and 4.1 after. These are amazingly equal, given the measurement imprecision. Remember, these angles were picked off a screenshot, where the unit of resolution is the pixel. For the resolution of our source frames and the short distances involved within the impact times, one pixel is worth perhaps a third of a degree.
  2. The 0.3 difference is not a constant, though the calculations treat it that way. The actual amount is the mid-impact direction minus the pre-impact direction. It is determined by a time, 225 microseconds, which is half the duration of impact. So the actual number depends on club head speed, swing plane and swing radius, but always comes out close to 0.3-0.4.

Last modified - Feb 1, 2014