Design Notes - Golf Physics p2

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Power
Impact
Launch
Ball Flight
Myths

What Happens at Impact

I'd like to thank Bernie Baymiller for the strobe pictures of impact. Bernie's father was the director of R&D for Spalding Golf in the 1940s, which is where and when the pictures were taken.

It's probably appropriate to begin the discussion of impact with a high-speed strobe photo of impact. So here's one -- perhaps the earliest ever taken. It was taken by Harold E "Doc" Edgerton of MIT, the inventor of the strobe flash.

Here are some facts about the impact shown here:

The ball is a Spalding Dot 100-compression balata covered wound ball. The club is a wooden driver with a 12-degree loft.
The total duration of this impact is 0.0004 seconds; that's 4/10,000 of a second, or 0.4 millisecond.
According to the caption supplied by Bernie, the clubhead moves 0.35" during impact, about a third of an inch. (My calculations suggest it is longer -- probably about an inch.)
The ball leaves the clubhead at 238 feet per second (162 mph). It is consistent with a clubhead speed of 120 mph, with the 1940s clubhead and ball in the picture. That's quite a high swing speed for that era, but achievable.
The 1.68" ball diameter compressed to 1.56" on the clubface, and elongated to 1.78" as it left the face.

My usual rules of thumb for impact are pretty similar to the collision described in the Spalding pictures:

Impact lasts no more than a half millisecond.
The clubhead moves less than an inch during this time.
The force between clubhead and ball can peak between 2000 and 3000 pounds. (It can average about 1900 pounds over the 0.4 millisec of impact. The peak is between 1.4 and 2 times the average force.)

So what is actually going on during that half-millisecond or so of impact? Here's the story, as described by Cochran & Stobbs -- and in somewhat more detail by Gobush:

In the first microsecond of impact, we have the irresistable force meeting the immovable object. The ball has to react somehow to the momentum of the clubhead, and the least-energy way it can react is by moving up the lofted clubface. Initially it slides up the clubface, because there isn't enough friction yet between clubface and ball. But all that is about to change very quickly.
The ball cannot acclerate to an upwards velocity in no time at all; that would require infinite acceleration, which requires infinite force. So it can't get completely out of the way of the clubhead just by sliding upwards. It begins to compress on the clubface, which creates a force between the clubface and the ball. If you don't think this is a large force, just try to compress a golf ball by 30% of its diameter using your fingers. Not even close! OK, use a vise; it's still very hard to apply that much compression force. Remember, this is a force that averages almost 2000 pounds during impact, and can easily peak around 3000 pounds. This force of compression does two important things:

It begins to accelerate the ball with a horizontal component, not just the vertical motion up the clubface.
It creates a lot of friction between ball and clubface. So, instead of sliding up the clubface, the ball begins to roll instead.

The ball continues accelerating upwards (due to the loft) and horizontally (due to the compressive force). The sliding has turned completely into roll, so the upwards acceleration increases the speed of roll. At some point, the momentum absorbed from the clubhead through acceleration has the ball moving faster than the clubhead. In other words, the elastic rebound of the ball's acceleration allows the ball to release from the clubhead. At this instant, its launch conditions are determined.

Launch Conditions

The term "launch conditions" refers to the ball speed, the ball direction, and the ball spin -- all taken the instant the ball releases from the clubface. Here is how impact conditions (things like clubhead speed, clubhead mass, loft, and other less important parameters) affect the launch conditions.

Ball speed

In a perfect collision, the ball speed would be given by the simple equation:

V_ball = V_clubhead 2
1 + m/M

Where m is the ball mass and M is the clubhead mass.

But in real life, the collision is not so perfect. There is some loss of speed for a variety of reasons. Here are the major reasons, and for each a factor to correct for it.

Energy loss due to compression of the ball - As the ball compresses on the clubface, and as it rolls in that compressed state, it loses energy to internal friction. (In fact, it's a USGA rule that it must lose a certain amount of energy, or the ball will not be approved as conforming.) In a collision, energy lost to internal friction is accounted for by a factor called the "coefficient of restitution", the fraction of velocity left after the loss of energy. Golf literature refers to this as COR, and physicists and engineers call it e in equations. In order to correct for this, replace the "2" in the ball speed equation by (1+e). Some interesting values of the coefficient of restitution are:
- 0.78 for the typical club today.
- 0.83 for drivers today, which get a little more restitution from flexible, springy faces. For more information on this, see my article on hard clubfaces.
- 0.67 in some of the older literature. When Cochran & Stobbs wrote their book, clubs and especially golf balls were not as "live" as today.
Glancing blow due to loft - The "perfect" collision of the equation above assumes that the impact is at right angles to the clubface. This implies not only a "squared-up" clubface, but also a zero-loft club. I don't know anybody who plays with zero-loft clubs, so we better have some way of accounting for oblique hits. Apparently different engineers estimate this loss differently. The diagram at the right shows four different estimates of ball speed loss due to loft:

Cochran & Stobbs, in Figure 23:5 of their book, present the effects of loft on launch conditions for lofts of 0°, 10° (a driver), 30° (a 5-iron), and 45° (a 9-iron). They give no indication if the data is experimental or computed.
The two trajectory programs I use, one from Tom Wishon and the other from Max Dupilka, compute launch conditions from collision conditions. I used those programs to compute the launch conditions for the four lofts in Cochran & Stobbs. The programs did not agree with each other, nor with the book's data.
I also computed the launch conditions using a simple formula. Just multiply the original ball speed by the cosine of the loft angle. That tracks the Dupilka program's output rather well, and lies between Wishon's and C&S' estimates.

Off-center impact - If you don't hit the sweet spot of the clubface, you will lose distance (ball speed). How much? I have seen a rough-and-ready estimator that says you lose 7% of your distance for each half-inch from the sweet spot. This is suitable for a first guesstimate, but isn't really correct. Two reasons it can't be correct we can deduce from an article by Howard Butler in Clubmaker magazine, dealing with clubhead moment of inertia:

The loss depends on the moment of inertia of the clubhead. Butler reports that high-MOI heads have 50% better resistance to loss of ball speed than non-peripherally-weighted heads. But the 7% rule includes nothing to account for MOI.
The loss is not linear -- that is, a constant percentage per half-inch. It is "square law"; if you double the amount you miss the sweet spot, you quadruple the loss of ball speed.

So, after accounting for all these losses, the initial ball speed is more like:

V_ball = V_clubhead 1 + e
1 + m/M cos(loft) * (1 - 0.14*miss)

where miss is the amount (in inches) by which the sweet spot is missed. That last factor is the most suspect in the formula, but is halfway decent for a first estimate if you need to account for off-center hits.

Direction of the ball

Let's take another look at the release of the ball from the clubface. The picture shows the direction and the spin of the ball (the red arrows), along with two other important directions: the direction the clubface is pointing (blue arrow) and the direction the club is moving (green arrow).

The direction of the ball -- called the launch angle -- is always between the arrows, and is almost always closer to the clubface direction. If there were no friction involved, then the ball would slide up the face and release in exactly the direction the clubface is pointing. But friction causes the ball to roll on the clubface instead of slide. The upwards motion of the ball is used to get the ball spinning. Since the ball has a moment of inertia, it takes some torque (force at the edge of the ball) to make it spin. That force comes out of the upwards acceleration of the ball, so it takes off a little lower than the clubface is pointing.

How much lower? The equation is a complicated, but not nearly as ambiguous as ball speed was. The three references (C&S, Wishon, and Dupilka) agree on the numbers for launch angle.

For very small lofts, the direction of the ball is nearly the same as the direction of the clubface. Put another way, the launch angle is the same as the loft.

As the figure shows, the launch angle becomes a smaller fraction of the loft as the loft increases.

For typical driver lofts, the launch angle is about 88% of the loft. For a 10° driver, the launch angle is just under 9°.
At about 20° of loft (the beginning of the irons) the launch angle is down to about 80% of the loft (the yellow dotted line). For lofts in this range, you can think about an 80:20 rule; the ball is 80% to the clubface direction and 20% to the clubhead path.
In the area of the wedges, the launch angle is still more than 60% of the loft. So, for all reasonable golf clubs, the ball's initial direction is closer to the clubface direction than to the clubhead path.

While the equations to come up with launch angle are complex, there is a simple formula that fits the curve very well. ("Fits the curve" means that the equation has nothing to do with the physics of the situation, but it happens to give the same result for all practical purposes.) This formula is:

LaunchAngle = Loft * (0.96 - 0.0071*Loft)

It gives results within a tenth of a degree up to 40° of loft, and stays within about a degree up to 60° of loft.

The same physics works in the horizontal direction. If the clubface isn't square to the path, the ball takes off between the two directions, and much closer to the clubface direction.

Since the loft is probably a bigger angular difference than the horizontal lack of square, you can use the loft to set the percentage difference of direction. For instance, the graph above shows that a 12° driver has about 85% conversion of loft to launch angle. So the sidspin due to a few degrees of non-squareness is also probably 85% in the clubface direction and only 15% in the clubhead path direction.

Spin

We've already seen the basics for creating spin. The ball compresses on the clubface, creating a lot of friction between the ball and the clubface. So the ball can't slide up the clubface any more; it has to roll. By the time it releases from the clubface, the roll has produced some number or RPM, which continues as the initial spin.

How much RPM? It is proportional to:

A decreasing function of the moment of inertia of the ball itself. Remember, MOI resists rotation, so the higher the MOI the lower the spin.
An increasing function of loft. Higher loft means more spin, all other things being equal.
An increasing function of clubhead speed. Harder impact means more spin.
A decreasing function of the internal friction of the ball, specifically its resistance to rolling while compressed.

When you factor all these things in, and remember that golf balls these days are rather similar in their gross physical properties, you get a formula:

Spin = 160 * V_clubhead * sin(loft)

Where the spin is in RPM, the velocity in MPH, and the loft in degrees.

Again, we can look at our three references for launch conditions. Their estimates of loft follow fairly similar -- though not identical -- paths. Our estimator based on the sine function tracks them well. (I'm not sure the Cochran and Stobbs numbers for spin were very carefully measured or calculated. They were rather casually listed as 60, 120, and 180 revolutions per second, as if the lofts were evenly spaced -- but they were not.)

There are a few more topics to consider about spin: grooves and gear effect.

Grooves do nothing for spin if there is clean, dry contact between the clubhead and the ball. Yes, I know this is counterintuitive; but it's true. So what about all the buzz you hear about the spin produced by square grooves?

For a little more insight into how and why square grooves help, let's think about automobile tires for a moment. Why do tires have tread patterns? To grip the road? Nope! If that were the case, then why would racing cars on a dry track wear "racing slicks", tires without any tread at all? The purpose of slicks is to have as much surface as possible in contact with the road. That, not a fancy tread pattern, will maximize traction. In fact, a tread pattern reduces the area of rubber in contact with the road.

So why do we have deep tread patterns on our non-racing tires? Because we don't always drive on a dry road. For all-weather use, you need a tread pattern to prevent hydroplaning, which is a complete loss of traction when the space between the tire and the road is filled with lubricating water. The grooves in the tread give a place for water to be channeled away, so rubber remains in contact with the road. If you look in a racing tire catalog, you'll see the different categories racing slicks and racing wets; even racers don't use slicks when the track is wet.

So how does this apply to golf clubs? It's a fact that grooves -- square or otherwise -- have little effect on spin for a fairway hit or a tee hit. Spin comes from the conversion of sliding to rolling as the compressed ball moves up the clubface. A ball compressed on the clubface by hundreds or even thousands of pounds of force can hardly get more friction than it already has, even if there are no grooves at all. (A study reported in Cochran & Stobbs gives more detail on this.)

But suppose there's some nice juicy grass between the clubface and the ball. It may provide enough lubrication to allow the ball to "hydroplane" on a smooth clubface. Like the tread on a tire, the grooves provide somewhere for the lubricant to be channeled away and allow steel to be in intimate contact with the ball.

So why are square grooves more effective than V-grooves in channeling away slime? The square grooves have a bigger volume, as you can see from the figure above. Remember, the USGA and R&A have rules setting the maximum width and depth of grooves. For the same width and depth, the square grooves have twice the volume; it's simple geometry. So you can channel away more lubricant with square grooves than with V-grooves. You may still lose some spin, but you lose less with square grooves.

I'm sure you've heard TV announcers talk about pros "hitting a flyer" from the rough. Now you can understand what's going on. The grass between clubface and ball lubricates the contact, allowing more sliding to take place. The results are:

Less spin, of course.
Higher launch angle! Because of the reduced friction on the face, the ball accelerates upwards more -- so it takes off much closer to the direction the clubface is pointing.

The combination of higher launch and lower spin allows the ball to fly farther -- a flyer. As a "bonus" (but one the player usually doesn't want) the lower spin means it will roll farther after it has already flown the green.

Gear effect is sidespin which is the result of an off-center hit with a club whose center of gravity is well back from the clubface. Without both these conditions, gear effect does not happen.

Let's see what causes gear effect. In the picture at the right, we have two off-center impacts, one on an iron and the other on a driver. Both are toe impacts, which means it is to the toe side of the center of gravity of the clubhead. (The CG is denoted by the four-quadrant black-and-white circle; it's a pretty common notation for CG.) What does Newton say about such an impact? The CG wants to continue moving forward in a straight line, but there is a force on the clubhead that is off that line. That creates a torque that wants to twist the club. The result is that the CG keeps moving forward, but the club rotates around the CG in a clockwise direction (red arrows).

The CG of the iron is close to the clubface. So, where the clubface and ball meet, this rotation (the red arrow) consists of the clubface "falling away" from the ball. This results in loss of distance (the momentum transfer is not as complete as it should have been), and perhaps the ball flying somwhat to the right as the face opens. But there isn't any special effect on spin.

The driver is a completely different story. Its CG is well behind the clubface. When the driver head rotates around its CG, the whole face of the club moves sideways. Look at the direction of the red arrow where the clubface and ball meet; it is mostly parallel to the clubface, with only a bit of "falling away".

So the club's face is moving to the right while the ball is compressed on it. The result is that the ball starts to rotate so its surface doesn't slide along the clubface; remember it's compressed so sliding is difficult. This rotation is the blue arrow in the picture. If the clubhead is rotating clockwise (as in the picture), then the ball rotates counter-clockwise. It's as if the clubhead and ball were a pair of gears, with their teeth meshing where they meet.

That's why a toe hit with a driver tends to hook. For all the same reasons, a heel hit with a driver tends to slice. You don't have this effect with an iron.

Last modified Jan 7, 2008