Hi Mr. Geoff Mangum,

I found your site and i found it very helpful. I am researching how physics
applies to golf for my research paper. In ur science section of ur site, i
was wondering if u could send me the following topics:

Impact of Physics- transfer of momentum/sweetspot
face angle orientation
putter sweetspot path
path relation to ball sweetspot

Ball-Turf Physics- gravity
skid, roll, decau
"true roll"
surface irregularities and grain

thank you very much,
Rachel

Dear Rachel,

Physics applies to golf in a number of ways, naturally. The main way is in how the club and the ball interact in a swing and hit. Then there is the physics of how the ball moves through the air, interacts with the ground (fairway, rough, hazard, or green), and interacts with the hole. You can also think about the physics of the human body in a golf swing.

I. Club-ball interactions

The main book for this is C.B. Daish, The Physics of Ball Games. He also wrote Learn Science through Ball Games (NY: Sterling Press, 1972), also covering the golf swing and flight of the ball through air. Here is some info about the second book.

The main idea is that the energy (momentum) of the club is transferred to the ball by the clubhead-ball collision. The energy of the clubhead is the mass of the clubhead times the velocity (speed + direction). The heavier the head, the more the energy for a given clubhead velocity. But there is a point where the head gets too heavy to move, and the velocity starts to drop off. Most DRIVER clubheads are near 7 ounces (about 28 grams per ounce, so about 200 grams), and most drivers in total (clubhead, shaft, grip) are around 11-12 ounces. A typical adult male clubhead velocity is about 100 mph at the bottom of the swing. Tiger Woods gets the clubhead up to 125-140 mph. Obviously, the faster the swing at impact, the more energy that transfers. The golfer has to balance clubhead mass against his ability to generate clubhead speed for the best mix for energy.

The typical golf ball itself weighs 1.62 ounces. Its velocity is zero to start with, so its momentum is zero too (mass times velocity).

The physicist looks at the golf swing at two precise moments in time to figure out how the energy/momentum transfers from club to ball -- just before impact, and just after impact.

Just before: The clubhead has all the energy (mass x velocity)
Just after: The clubhead has some and the ball has some.

The BIG IDEA from theory to help figure this out is that the TOTAL energy in the ball and clubhead just before is the SAME as the TOTAL energy in clubhead and ball just after. Nothing funny happens to the energy, and it is all accounted for by physics. This is called the Law of Conservation of Energy, but it just means nothing funny happens to the energy.

To use this idea, set up formula for before and after for the Total Energy, and then put the formulas together, this way:

Just before: Clubhead Energy (mass x velocity) + Ball Energy (mass x velocity, which is zero) = Total Energy
Just after: Clubhead Energy (mass x velocity) + Ball Energy (mass x velocity) = Total Energy
SO THAT:

Clubhead Energy Before + Ball Energy Before (0) = Clubhead Energy After + Ball Energy After

The one you need to know is Ball Energy After. To fugure that out, you have to know the mass of the clubhead and its velocity before impact and after impact. Typically, the mass is 200 grams and the velocity before is 100 mph before and after, the clubhead still has some velocity but a lot less (maybe 60 mph).

UNITS: You have to use the same system of units for physics formulas, and unfortunately the English system is not very convenient (inches, feet, pounds) and the metric System is a lot easier (meters, grams). So you have to convert mph to meters/second. We know that 1 meter has 39.37 inches, and that 1 hour has 60 minutes with 60 seconds each or 60*60 seconds (3,600 seconds in 1 hour). First convert mph to inches per second. 1 mph is 5,280 feet per hour, and in terms of inches is 5,280*12, or 63,360 inches per hour. Covert this to inches per second: 63,360/3,600 = 17.6 inches per second. So, 1 mph is the same as 17.6 inches per second. That means that 100 mph is the same as 17.6*100 or 1760 inches per second. How many meters are covered vy 1760 inches? 1760/39.37 = 44.70 meters per second. So 100 mph is the same as 44.70 meters per second. (A car traveling this fast covers 44.70 meters of roadway every single second. How many feet is that? 1760 inches / 12 inches per foot = 146.7 feet every second!)

How about 60 mph? 17.6 inches/sec/1 mph * 60 mph = 1056 inches / sec. 1056 in/sec divided by 39.37 inches/meter = 26.82 meters per second. (How many feet? 1056/12 = 88 feet every second is covered by a car traveling at 60 mph).

So, back to the formulas:

Before: Total Energy = 200 grams * velocity (use kilograms instead of grams, so use 0.2 kg)... 0.2 kg * 44.7 m/s = 8.94 units of energy (kg-m/s).

After: Ball Energy + Clubhead Energy = 8.94 kg-m/s (by the Conservation Law)

So, Ball Energy = 8.94 - Clubhead Energy After
So, Ball Energy = 8.94 - (0.2 kg * 26.82 m/s)
So, Ball Energy = 8.94 - 5.36
So, Ball Energy = 3.58 kg-m/s

Hoiw FAST is that ball moving?

Ball Energy = mass * velocity

Ball mass = 1.62 ounces. Convert to kilograms: 1 ounce = 28 grams, so 1.62 ounces = 1.62 * 28 = 45.36 grams. I kg = 1000 grams and 1 gram = 0.001 kg, so 45.36 grams */ 1000 g/kg = 0.04536 kg (round to 0.045 kg).

Ball Energy After = 3.58 kg-m/s = mass * velocity = 0.045 kg * Velocity m/s
So, 3.58 kg-m/s = 0.045 kg * Velocity m/s
So, Velocity m/s = 3.58 kg-m/s divided by 0.045 kg
So, Velocity m/s = 79.55 m/s

How fast is that in mph? 79.55 meters * 39.37 inches / meter = 3132 inches; this is 3132/12 feet, or 261 feet per second. There are 3,600 seconds in each hour, so in feet/hour, this is 261*3600 = 939630 feet per hour. There are 5280 feet in each mile, so in mph this is 939630/5280 = 118 mph!

This means that swinging a 200 gram clubhead at 100 mph into a 45 gram golfball on a tee so that after impact the club slows to 60 mph, the ball shoots off into the air at 118 mph! Fore! The ball travels through the air and is slowed by having to knock all the air molecules out of the way, plus the earth's gravity makes it drop downward all the while, so its flight path looks like a rainbow, at the end of which it bounces on the ground a couple of times and rolls to a stop. The AVERAGE velocity during the whole flight is something like halfway between the top speed (118 mph) and the bottom speed (0 mph), so the Average speed would be about 59 mph. Each mph is the same as 1.47 feet per second, so 59*1.47 = 86.7 feet each second of travel (on average). If the ball stays in the air about 6 seconds, it will travel 520.4 feet. That's a drive of 520.4/3 yards, or 175.3 yards. That's probably about right for a lot of people.

At Golf Stores, they have a machine that can measure how fast the clubhead is moving BEFORE impact, but we are really guessing about how fast it is going JUST AFTER impact, so all these numbers require precisie scientific measurement. But you can still gets some numbers on the golf course, and work backwards with the physics to try to figure out other numbers. For example, a man hits a golf ball 200 yards with a 100 mph swing speed and it flew for 6 seconds. How fast was the clubhead moving AFTER impact? 200 yards = 600 feet = 600/6 seconds = 100 feet per second average ball speed, so top speed just after impact of ball is 200 feet per second. 1.47 feet/sec for each mph, so 200/1.47 = 136 mph ball speed just after impact. Total Energy before = Total Energy after, so

Ball Energy after + Club Energy after = Ball Energy before (0) + Club Energy before

136 mph * 0.045 kg + Club speed after * 0.2 kg = 0 + 100 mph * 0.2 kg
6.12 mph-kg + clubhead speed after *.2 kg = 20 mph-kg
Clubhead speed after * 0.2 kg = 20 - 6.12 mph-kg
Clubhead speed after * 0.2 kg = 13.88 mph-kg
Clubhead speed after = 13.88 mph-kg / 0.2 kg
Clubhead speed after = 69.4 mph

This is a little faster than we estimated (60 mph), since the ball went a little further (200 yards vs 173 yards). Nothing funny happens to the energy. The lesson is that if you know the mass of the clubhead and your swing speed, you just count how long the ball is in the air and see how far it goes. These two numbers plus the formulas allow you to figure out the average speed of the ball, to top speed of the ball after impact, and the clubhead speed after impact. Neat!

The same series of calculations have to be done for each club and each swing speed. There are no more than 14 clubs in a set, including the putter, and they all have different masses and swing speeds.

VELOCITY- the concept of velocity is BOTH speed and direction. All along, we have been assuming a STRAIGHT swing with a straight clubface and a resulting straight ball flight. Unfortunately, in golf, swings aren't always straight, or the clubface is a little twisted, or both. Then, all the formulas have to use only the straight part of the hit, for the most part. The straight part of the hit is just that energy that is moving on the straight line of the blow, OR the straight line of the ball flight. They won't be the same in a cockeyed hit with the clubface a little twisted. When the clubface is really square, pointed the same way the swing is moving the clubhead, the ball will go the same direction. But if the face is twisted different from the direction of the clubhead movement, some of the energy of the clubhead is sort of wasted. The ball doesn't go the way the face is pointing or the way the clubhead is moving, but in between the two. To figure out precisely where the ball will go requires separateing the forces into "components" sideways and straight ahead. This makes a triangle (base = left or right, height = straight) of forces, and the one that RESULTS in the ball direction is in between these two directions. The way to see this is to imagine the triangle, and then make a duplicate triangle, flip it, put the two together in a rectangle, and draw a diagonal from corner to corner. The diagonal is the RESULTING direction of the ball flight, adding the two separate "components" back together. If you need to know more about the sidewise and the straight ahead parts, you have to use a little trigonometry (sine, cosine, and tangent) to get the relative LENGHTS of these parts of the forces. The LENGTH is the speed part of velocity (speed plus direction) and the ANGLES tell you direction. Sine, cosine, and tangent just let you start with a known angle and one side to find the other side or the hypoteneus (diagnonal), or start with two sides and find an angle, and so on. That's pretty advanced.

Another assumption has been that the ball and club smack each other and don't change shape from the impact. In fact, they both change some -- the club (metal) practically none, but the ball (plastic and rubber) quite a bit. In slow motion, you can see a club "smushing" a golfball, so that nearly one-third of the ball gets smushed flat by the club. What really happens is that the clubface is angle up a bit (more and more from driver to pitching wedge through the set of clubs) -- straight but up some. When the club smushes the back of the ball flat, it also smushes it a little upward. The ball then rebounds back to shape (from the physics of the ball design) AND slides up the clubface a bit. As it rebounds, it is accepting the energy from the club (and the club is slowing down as it loses or gives up energy to the ball) and the ball "shoots" off the clubface in a straight but upward direction. It also is spinning with backspin, so if you watched from directly opposite the golfer as he swings, the bottom of the ball moves up forward and back over the top towards the golfer.

For all clubs but the putter, the ball flies through the air, bounces, and rolls to a stop. The physics of balls bouncing on the ground take into account how hard the ground is and the angle the ball comes in at and how fast the ball is going when it hits.

II. Ball Flight through the Air

The physics here is all about the ball blasting through all the tiny air molecules. The ball is spinning, and if hit straight, all the spin is backspin in the plane of the flight and none is sidespin. This spin affects how the ball cuts through the air. Also, the dimples on the ball affect the ball-air interaction. Basically, hitting all the air molecules is slowing the ball down (the same way the club hitting the ball slowed the club down), but its pretty gradual because the air molecules are so small (light mass) compared to the ball. The air "flows" around the ball. This means the ball cuts or pushes through the air gas cloud like a bullet fired into water. The air is pushed apart and then behind the ball the general air pressure pushes the air molecules back. There is a bit of a vacuum behind the ball because the ball is dirturbing the generally equalized air pressure as it flies. This vacuum pocket behind the ball is called "drag" because it literally sucks on the ball and slows it down a bit. How do dimples affect this situation? By lessening the drag vacuum so the ball will go farther. How? By chopping up the airflow as it passes around the ball, the airflow at the back of the ball is more mixed up and directed some into the blank area behind the ball. This means SOME of the air pressure is kept close to the ball and there is less of a suck back there.

What does the backspin do? The backspin means the bottom of the ball is hitting the air faster and harder than the top of the ball. In other words, the "pressure" of the ball-air collissions is greater on the bottom and less on the top. This is the same "lift" that an airplane wing creates in cutting through the air, so the ball rises because it spins. The loft of the club starts the ball out at a certain angle upward but spin makes it go even higher upward. If you launch a driver shot at 17 degree (hitting with a 12 degree loft 5 degrees up through the ball), the ball starts out at 17 degreees upward but is also spinning backwards, so it "climbs" an even steeper angle up (maybe 20 degree?).

There is an ideal or optimum combination of launch angle, spin, and length of total flight. It's usually somewhere around 17 degrees. So if you have a 12 degree loft on your driver clubface, hit slightly up through the ball on the tee for maximum distance.

With other clubs, the loft changes. The MOST loft is the wedge. Most wedges have about 56 degrees (while a driver has only 8 to 12 degrees). Combined with backspin, the wedge sends the ball WAY high and not too far out. So for the whole set of clubs, each club is suited to a certain distance shot, as may be called for when playing the course. AND each club has a typical shape of the ball flight -- differently shaped rainbows for different clubs -- and the shapes get shorter and higher going from driver to wedge. Or lower and longer going from wedge to driver.

What does sidespin do? Sidespin comes from hitting the ball with the face twisted and aimed differently than the direction of the clubhead movement. The club smushes a bit of the side of the ball and then when the ball rebaounds, it has a little side spin PLUS a little back spin. These two "components" add up to a RESULTING spin that is tiled out of the vertical plane of the ball flight (just like we did above for the clubface). This spin weakens the ball-air pressure on that side of the ball -- right spin lessens the pressure to the right. So the ball has "lift" off to the side and flies off to the side, depending upon how much sidespin is in the total mix. A slice is a lot of right spin in the mix. A hook is a lot of left spin in the mix. What causes slice spin? Twisting the face to the right in relation to the direction of the clubhead movement. This is said to "open" the clubface. If you swing straight with an open face, you ball spins right and slices. If you swing straight with a "closed" face twisted left, the ball spins left and hooks.

Another assumption we have been making is that the clubface and ball make contact like two ideal points. Two billiard balls can hit head-on or with a glancing blow. We have been assuming a head-on impact. This is all about the center of gravity of the clubhead and ball. The ball is easy - the center of gravity is the center of the sphere. The clubhead is hard. The center of gravity is that one point in the total shape of the clubhead where the different distributions of mass in the weird shape all come together for balance. A round ball with a uniform material throughout has a center of gravity (CG) in the center, but a driver clubhead is more like a bigger ball cut in half and flattened mostly and then stuck on the end of a stick. Depending upon the design of the driver, the CG may be halfway up from bottom to top and halfway back from clubface to back, but not necessarily. In the blow, if the clubhead's center of gravity is moving BELOW the CG of the ball, it will make the club twist upward through impact (the club CG sort of wraps beneath and up around the ball CG). This makes the ball go higher because it alters the loft and launch angle. Golfers who want to hit the ball higher with any club add lead tape to the bottom of the club. To hit the ball lower with that club, do the opposite -- on the back of the club but nearest the top, add some lead tape. The tape moves the CG higher or lower.

The same happens with a clubhead CG off to the side as it moves through the ball. Club CG to the outside of the ball's CG makes the clubface twist around to the inside (left), and this creates hook spin. A club CG inside the ball's CG creates slice spin. This is separate from clubface twist, so you can have a straight swing and a straight clubface but have the clubface CG inside the ball center as it goes through impact and STILL get some slice! Tough game!

Drivers are designed so that the clubface has a "bulge" in front of the club's CG. This way, if you contact the ball with the CG inside, the contact is more glancing than it would be with a flat face, and this reduces the sidespin so the ball doesn't slice so much. It also reduces hook spin the same way.

III. Green-ball interactions.

In putting, the roll of the ball across the green is affected by gravity and the ball hitting the grass blades as it rolls. If the green surface is tilted, then gravity is free to influence the direction of the ball. It is like driving a car straight on a level highway. If the green is tilted, it is as if the steering wheel of the car is given a slight turn away from straight. This constant influence will make the car curve off the road all right, but the exact shape of the curve of the car depends on how fast the car is rolling. The same is true of a ball rolling on a tilted green. Daish covers this pretty well.

The force of the ball is figured just as it is for other clubs, so this tells you that the putterhead speed and mass determines how fast the ball will start out after impact. (And the face angle and direction of the blow matter as well, and so does the CG or sweetspot of the putterhead and ball.) But after the ball starts out, here's what happens:

A "roll" of a ball is not the same as its forward motion. When you first punch a ball with a putter, the ball skids over the grass because it is not rolling yet. The grass friction on the bottom makes the ball start to spin and roll (slows the bottom of the ball down and the top of the ball spins forward) and it takes a little bit before the spinning / rolling matches the forward skidding. When the circumference of the ball rolls along over the grass without skidding, then the grass has added enough spin to the ball that the roll matches the forward motion. When it does, the skidding no longer happens and the ball just rolls. The shaggier the green, the quicker this skid phase is over because the extra grass friction spins the ball up to speed quicker. On really "slick" greens, the skid phase is longer and matters less. Typically, the skid lasts about one seventh of the total putt length. A seven foot putt has a typical skid of one foot, and a 14 foot putt has a skid of about 2 feet.

Also, when the ball is at rest to begin with, it's bottom sits nestled down in the grass probably all the way to the dirt, sort of cradled by the spongy grass blades. When struck with a putter, the ball moves and skids over grass ahead of it and this makes it rise up a bit (the grass blades get shoved down and create a slight incline plane that perpetually lifts the ball up a bit). As this lifting continues and as skidding becomes pure roll, the ball enjoys a state of least interference from the grass and gets a lot of ground covered nicely. Eventually, though, the constant friction of the grass takes its toll and the ball reaches a point where the "inclined plane" effect goes away and the grass is seriously in the way again. This is the "decay" phase of the putt where the ball slows down quicker. For any given green with grass conditions of a certain sort, this phase will set in whenever the ball slows to one specific speed. Then the ball slows quickly to a stop. The interesting point is that on one green the decay phase is always about the same length of roll. A typical length for a typical green is about 1 foot.

So, to putter better with this knowledge, you want to know if you can influence the skid and you want to know what to do about the decay. yes, you can influence the skid by making it more or less, but what difference does it make? Many golfers believe that skidding increases the chance of the ball going off line by hitting something low in the grass. I don't know about that. At any rate, if you want to REDUCE skid you have to either start the ball with some overspin or increase the grass friction to get the spin up to speed. Hitting slightly up on the ball with the putter is thought to add overspin at the start and thus reduce skid. Hitting above the ball's equator is also thought to add overspin. In either event, you are lessening the energy transfer from club to ball unless there is sweespot-to-sweetspot head-on contact between the CG of the putter and the CG of the ball. It's a complicated matter, because less skid makes the ball end up going farther but not having sweetspot-to-sweetspot contact means the ball goes less far. It's a trade-off you have to think about and try out for best effect. Also, hitting up lessens grass friction and so tends to make any skid last a little longer, while hitting down across the top of the ball adds friction. Because of this, it would seem hitting down slightly across the top of the ball would be a little superior to hitting up. On the other hand, perhaps hitting up makes the "inclined plane" effect happen quicker. Also, hitting down makes the ball smush into the spongty turf and then rebound out and it might bounce off line. Hitting up also might "launch" the ball and make it hop along and go offline. Even if bouncing or hopping stays on line, this lessens the energy available for going straight, so these balls don't usually go as far (come up short). As I say, it's pretty complicated. Personally, I think you should just go with a level blow through the ball's sweetspot and not worry about skid. Most greens today are so "nice" that skid doesn't matter much.

IV. Ball-Hole interactions.

For a rolling ball to pass over the front lip of a cup and fall in, it has to drop at least half its diameter before it hits the back wall or rim of the cup. Otherwise, it will hot the rim on the bottom half of the ball below the equator and bounce out of the cup and keep going across the green. This means the key is HOW FAST IS THE BALL ROLLING AT THE FRONT LIP? Is it slow enough to permit gravity to make the ball drop one-half or more of its diameter?

To figure this out, get the numbers first. A golf hole is 4.25 inches wide and 4 or more inches deep, cylindical in shape. A ball is 1.68 inches in diameter, and one-half is 0.84 inches. The speed of the ball across the 4.25 inch center of the cup must be slow enough to allow gravity to pull the ball down 0.84 inches before the leading edge of the ball hits the back wall. Note that the actual distance across that is traveled acroiss the center of the cup is not the whole 4.25 inches, because half of the ball is already across before the bottom of the ball crosses the rim. So the real distance across the center is only 4.25 - 0.84, or 3.41 inches. So how slow must a ball move across 3.41 inches to allow it to drop 0.84 inches? This is the same as asking how long does it take for gravity to make a ball drop 0.84 inches.

Galileo on the Leaning Tower of Pisa dropped a ball and discovered that gravity pulls on a ball so that it gains speed at the rate of 32 feet per second every second. Starting at zero feet per second when he first lets go, at the end of one second gravity has sped the ball up to a velocity of 32 feet per second. At the end of two seconds, the ball has sped up to 64 feet per second, and so on faster and faster until the ball hits the earth. (A 20-pound bowling ball dropped from the Empire State building will hit the sidewalk going so fast it will explode and blow a crater in the concrete.) At the end of the first second, how far has the ball dropped? That depends on its average speed. The top speed is 32 feet per second, but the start speed was zero, so the average is 16 feet per second. At the end of one second at an average speed of 16 feet per second, the ball has dropped 16 feet. A golf ball would drop the same distance in one second, since the earth's gravity has the same effect regardless of the mass of an object. (Try droipping a quarter into your hand and then a penney -- they both fall at EXACTLY the same rate even though the don't weigh the same.)

Our question is how many fractions of a second is it for gravity to move the ball 0.84 inches. The math expression for what we have been discussing is simple: Distance = speed * time. The speed is the Average Speed from gravity (top speed - start speed / 2). Since start speed is zero, average speed is just 1/2 top speed. The top speed from gravity depends only on how long the object has been falling, since gravity gives the object a constant uniform acceleration (32 feet per second each second). Top speed at 2 seconds = 64; at 3 = 96 etc. So top speed = gravity acceleration * time.

Back to our basic formula, Distance = speed * time, where speed is average speed, and average speed is 1/2 * top speed from gravity, and this in turn is gravity acceleration * time, the resulting formula is Distance = [1/2 gravity accleration * Time] * Time. We want to know Time.

Distance = [1/2 Gravity acceleration] Time Time
Distance = 1/2 32 feet per sec per sec Time (squared) seconds-seconds
Convert 0.84 inches to feet = 0.84/12 = 0.07 feet
Convert 32 feet per second / second to inches per second / second, 32*12 = 384
1/2 * 384 = 192
Time (squared) = 0.84 inches / 192 inches per sec per sec
Sqaure Root of Time (squared) = Time, so
Time = Square Root of [0.84/192]
Time = Square Root of 0.00438
Time = about 0.067 seconds (7 one-hundreths of a second, less than 1/10th of a second).

Now, if Distance = Speed * Time, and the distance across is 3.41 inches and the time is 0.07 seconds, the Ball's Speed at the rim has to be 3.41 in / 0.067 sec OR SLOWER, and that speed is about 51 inches per second. Any ball rolling faster than this has NO CHANCE of falling into the cup even if it crosses the widest possible part of the cup. How fast is that? A ball rolls one full circumference per roll, and the circumference of a ball is Pi* Diameter or 3.14 * 1.68 = 5.28 inches around. So each roll, the ball travels 5.28 inches. A ball rolling at 2 revolutions per second speed covers 10.56 inches each second. A ball rolling 51 inches per second is the same as 51 / 5.28 = about 9 revolutions per second. Anything faster than this has no chance.

To consider balls that don't go across the center of the cup, but cross a shorter path off to the side, just consider a line across the cup merely 2 inches from front rim to back rim. The ball only has 2 - 0.84 inches before it hits the back rim, or 1.16 inches total. 1.16 / 0.067 = 17.3 inches per second MAXIMUM SPEED, and 17.3 / 5.28 is 3.3 revolutions per second. So if you want a chance with most of the cup available, a good speed for the ball as it crosses the rim is around 3 revolutions per second OR LESS.

If a ball going 17 inches per second missed the cup altogether, on most greens it would probably stop about 1 to 2 feet past the hole. You can use this knowledge to advantage by trying to control your putts so that they usually stop a little past the hole, and not more than 2 feet past. I personally don't want to miss, but if I do I try to have the ball stop within 1 foot of the hole, so my speed is about 2 revolutions per second or under at the front rim.

Some other physics books that can help are

Jorgenson, The Physics of Golf (all about the full swing)
Pelz, Putt Like the Pros (some of the physics of putting)
Templeton, Vector Putting (about gravity on the green and grass friction)
Cochran and Stobbs, Search for the Perfect Swing (mostly full swing but some about phases of a putt)

These books might be hard to get, but a refernce librarian can track them down for you using Interlibrary Loan.

There is also an article or two in the Physics journals about how a ball and hole interact. They are very technical mathematically. If you want those references, let me know.

Cheers!

Geoff Mangum
The PuttingZone
http://puttingzone.com>
The Future of Putting Now - elite instruction, comprehensive resources.

PS - if you haven't already, please join my free PuttingZone Mail List and get the good stuff first!

Hi Mr. Geoff Mangum,

Hello, my name is Rachel and for my research paper I am researching: How Physics Applies to Golf. I have spoken with you earlier this month and I was hoping if you could answer the follow interview questions for my paper. I know that you are very busy and all, however I would greatly appreciate your response to the questions below:

Dear Rachel,

Here are my responses:

1. Mr. Mangum, what is your occupation? How did you get started?

Putting theorist & instructor. Just decided 10 years ago to do it.

2. Is golf and Physics related? How?

Yes, physics describes the forces of nature, and in "classical" physics, the main forces studied are those involved in the collisions between objects and the movement of objects. In other words, how forces affect objects by changing their state of rest or motion, and how forces are transmitted (mainly through collisions). So, in golf, how the body is moved (biomechanics and kinetics) determines how the club is moved, and the design of the club determines how the forces of the club in motion are applied and transferred to the ball; the design of the ball determines how the energy of the force is received by the ball and how the ball interacts with air molecules as it flies; then the character of the ground interacts with the ball when gravity brings it back down to earth and it bounces; and the shape and size of the hole conditions how gravity is allowed to work as the ball crosses over the cup and hopefully falls in.

3. How does your occupation relate to golf and Physics?

Gotta know to teach.

4. In what ways does Physics apply to golf?

See 2.

5. In your opinion, is golf meant to be played/understood with Physics?

I'm not sure I would say "meant to be," because golf is meant to be played by people regardless of whether they understand physics. However, many people enjoy trying to understand things more deeply, and I'm one of them. To a certain extent, everyone has some basic or rudimentary understanding of how objects move and collide, as we all deal with this everyday. Most of this understanding is close enough to accurate physics, but the study of physics can explain a lot of little mysteries and can help you understand many many things about nature all at the same time, not just stuff about putting. So learning golf physics is just one way of learning some very interesting things about nature.

6. Is there just one swing that produces the greatest results?

Every swing is made by a different body, and every body is unique, even if they all fit within a certain range of sizes - so "no", there is not one unique swing. However, there is one set of physics laws that apply to everyone all the time, and for any given body to do its best, it will have to take advantage of the laws of physics. So, for all golfers, there are certain guiding principles for the swing that need to be observed. The top golfers just apply the laws the best, or are able physically (body shape and fitness, etc.) and mentally to perform golf skills with good physics.

7. Is there such a thing as a "Physically perfect" swing?

There is an ideal swing for each body on any given day with a certain club and ball and environment. It all works together, since it is all of nature we are describing in the golf swing. And it really differs a little from day to day. On cold days, there is a different "ideal" combination of equipment and swing (and even clothing that warms but does not hamper body movement). And cold golf balls don't go as far, because the ball is cold and harder but also because the air molecules are cold and slow and packed together. On the other hand, on a very cold day when the ground is very hard and the grass all dried and dead, the ball will land and bounce and run forever if you use a certain sort of swing.

8. Why do golf balls have dimples?

The dimples make the ball go farther through the air molecules. What happens is the ball jams its way through the crowd of tiny molecules in the air gas, and as it does, a "stream" of air is opened at the front of the ball, rushes around the ball, and closes back once the ball has passed. However, the air stream doesn't close back until a little gap behind the ball. This gap is then a vacuum, because the concentration of air here is very low, which means the pressure is low. A vacuum literally "sucks" on the back of the ball and slows it down. It's not that big of a sucking, but it's real steady and constant the whole time, and cumulatively it really takes a toll on the ball's total flight distance. It's kind of like coasting on a bike but dragging both feet in the dirt. The dimples "chop" the air stream up on the way by the ball and this makes the air close back behind the ball a little quicker. This reduces the gap size and thus the strength of the sucking. Cumulatively, a little reduction of this gap goes a long way to helping the ball get farther through all the air molecules. Golf companies use very expensive computers to design the shape and puzzle-pattern of fitting dimples on the sphere of a ball, so they have to know a lot about geometry and polygons. originally, back in golf history, dimples were discovered by caddies, who noticed that new ball were smooth but didn't work well until they got scuffed up from use. So they started putting little scuff and knife marks on the balls. Later, ball manufacturers just made the molds with a rough pattern.

9. How is the projectile of the golf ball related to where it was hit? (Fat, thin, square)

The energy transfer from club to ball depends upon whether the impact is "solid" or less than solid. A solid hit is one where the center of gravity of the club-n-motion passes straight through the ball's center of gravity during the brief instant of contact. This is sweetspot-to-sweetspot contact. Such a hit sends the ball the farthest. A "thin" hit is a glancing blow where the club's sweetspot does not move through the ball's sweetspot, but along a line off to the side. In this case, not all the energy of the club is transferred to the ball, and some is wasted. This shot does not go as far as you planned. Usually, with golf irons (pitching wedge through fairway woods, but not the driver or putter), the golfer wants to "take a divot" from the earth in the shot, but always just the right size divot. A "fat" shot results in way too much divot. A "fat" shot is one where the club chunks too much into the ground. When you "hit the big ball before the little ball" as golfers say, this collision between the club and earth transfers energy from the club to the earth, not to the ball. A "fat" shot is one where too much energy of the club-in-motion goes to the earth and there's only a little left to send the ball away, so it goes well short of what you planned (and you can break your club or your wrist - especially if you collide with a rock or a tree root).

10. How does a golf ball's flight relate to Physics?

See 8. Also, the energy of the club is the mass of the club times its velocity (speed in a certain direction). So, for a given club, the faster the clubhead is moving at impact, the farther the ball will go. And, the heavier the club for one swing speed, the farther the ball will go. The trick is to get the best combination for you body and ability. That's why golf clubs have to be specially "fitted" to an individual golfer, sort of like buying shoes.

In addition, the ball's flight through air molecules involves the physics of "lift" and drag." Lift comes from the ball's backspinning, since the loft angle on the driver face and other clubs sends the ball off with the bottom spinning forward and up and back over the top of the ball. Lift means the front bottom of the ball hits the air molecules harder than the top front of the ball, which is moving the same direction as the air. This creates higher pressure at the bottom of the ball and lower pressure
at the top, so the ball rises like a glider plane in addition to the upward angle of its launching. Also, if the ball is given sidespin at the start, this can make the ball go left or right off line (slice or hook) in addition to rising. Many golfers do this deliberately to control where the ball goes, but many more golfers get sidespin by accident. Drag is a pocket of low pressure behind the ball as it speeds through the air gas molecules like a bullet through water. The ball shoves the air out of the way, and the air stream flows around the spehere of the ball and then closes back behind the ball, but only after a little gap opens up back there. This is a low-pressure pocket that litterally "suck" constantly on the ball a little, and keeps the ball from going as far as it would otherwise. Dimples reduce drag, but don't eliminate it entirely, and some dimple patterns are more effective than others at this.

11. Is there an equation?

Yes. Drag depends on the cross-sectional area of the ball (the circle seen from the front), the density of the air (called "rho" from the Greek letter), the air speed of the ball (velocity in relation to air velocity - is there a tail wind or a head wind?), and a "coefficient" of drag having to do with the specific ball design. Drag isalso affected by spinning, but that's very complex. The typical mathematical expression of these relationships in a formula is: Drag = Coefficient of drag * Area (cross-section) * 1/2 Density of air * Air speed (squared). D=CA(1/2)rho*v*v. The Coefficient goes down as the speed goes up, anbd also goes down depending upon the dimple design. So, the faster you hit a ball, the less drag there is, and fast swingers like Tiger Woods get a little help from physics because they don't have to worry about as much drag as other golfers. That ain't fair! He's already better, but physics makes him even better yet!

The bigger the ball, the more the drag. The British used to use a smaller ball than the Americans (only 1.62 inches in diameter versus the American ball's 1.68 inches). Needless to say, they had an advantage over Americans in the British Open and the Ryder Cup matches.

In the mountains, like those in Colorado, the air is thinner, so density is less. So, in high elevations, the ball goes farther, since the drag is reduced by less dense air. Balls go 10% farther or more, especially on a warm clear day.

In the summer, the air is warmer, and a warm gas is a less dense gas. So, the ball goes farther in warm weather, unless humidity makes the air still dense. Incidentally, the temperature also affects the golf ball in terms of it's ability to rebound off the club when it gets smushed. When cold the ball doesn't rebound very quickly so it doesn't fly as far. (Don't keep your golf balls in the trunk overnight during cold seasons - it takes hours to get the ball warmed back up all the way to the center, and putting them in your warm pocket only warms up the thin cover.)

The formula for Lift is pretty much the same: L=CA*1/2rho*v*v. Here, C is the coefficient of lift. Lift gets bigger the faster the ball is going, so really powerful golfers use drivers without a lot of loft (say, an 8 degree driver, versus other golfers' 12 degree driver loft). The faster ball rises higher (and slices and hooks worse too). And lift gets bigger the faster the ball is backspinning. Wedges with 56 degrees of loft spin the ball a lot faster than a 12 degree driver, so wedge shots go very high compared to the distance of the shot, while drivers go longer than they go high. A wedge rainbow is very bowed up or humped in shape, while a driver shot is more flattened. Other clubs in between these two produce flight shapes in between these extremes that are graduated in a step-wise fashion.

Once you have Drag and Lift, you go back to the basic idea that how far something travels depends on its average speed and how long it is traveling. So, Distance = Speed * Time. This is how you decide how far a golf ball will fly. It's a little complicated by Drag, Lift, ball spin, air density, and initial launch angle, but basically these are details added into the basic notion that Distance = Speed * Time.

For more details, you can read Daisch, The Physics of Ball Games (1972) or Herman Erlichson, Maximum projectile range with drag and lift, with particular application to golf, American Journal of Physics, April 1983, vol 51, no. 4, pages 357-361.

12. How does putting relate to Physics?

Body biomechanics, ball-club collision, ball-ground interactions, and ball-hole interactions are all governed by classical physics laws and the laws of gravity.

13. Is there a specific equation to describe how putting relates to Physics?

No, there is a whole little set, depending upon what part you want to know about. For example, in the movement of the putter in the stroke, the laws governing how a pendulum moves apply, but most people don't follow these laws strictly, but try to add or subtract from the "natural" pendulum motion. What ends up mattering is just what does the golfer actually do. The formulas have to be adjusted for the specific situation: different golfers, different clubs, different balls, different greens, different weather. What most matters is knowing the BASIC laws of physics (conservation of energy and momentum, the three laws of Newton) and then knowing how in a scientific approach to understanding what is happening on a given occasion depends upon adjusting these laws to apply them to the case at hand. You end up having to know some numbers about your body, club, ball, and green. For example, you need to know how heavy the clubhead is, or at least use the formula to figure it out from some other things you know.

Even though there is a collection of formulae, a putt can be broken down into phases of 'flight" (Fd) at the very start, "skidding-rolling" (Sd) after that for a short part, and the "pure rolling" (Rd) the rest of the way. The total distance (D) a putt travels is then these three phases added together, D = Fd + Sd + Rd. There are separate formulae for each phase.

The Fd (in feet) = 2* Launch velocity (ft/sec) Squared * cosine of Elevation Angle of launch (in radians, not degrees) * sine of Elevation Angle of launch (in radians) / Accelerating force of gravity (in feet per second per second)

The Sd (in feet) = (Launch velocity * cosine Elevation Angle - [Acceleration of gravity*Coeeficient of friction of sliding over the grass on that green*Time of slide (secs)/2] * Time of slide (seconds)

The Time of sliding or skidding (in seconds) = (Velocity oif launch * cosine Elevation Angle of launch - Angular velocity at launch (in radians per second of spin, negative for backspin) * Radius of the ball (i.e., 0.84 inches / 12 inches per foot)) / (Coefficient of sliding friction* (Acceleration of gravity + Ball weight (i.e., 1.62 ounces) + Ball radius Squared / Moment of Inertia of the ball (in slug-feet Sqaured))

The Coefficient of sliding fricition depends upon the Stimpmeter speed of the green. Coefficient = .946 -.021*Stimpmeter reading (in feet). There is no unit for this coefficient.

The Rd (in feet) = .0477*Stimpmeter reading (in feet)*Velocity of ball at point skidding stops and becomes pure rolling, raised to the 1.595th power.

The Velocity when the ball stops skidding (in feet per second) = Velocity at launch *cosine Elevation Angle at launch - Acceleration of gravity Coefficient of sliding friction Time of sliding.

Pretty complex, huh? Basically, you have to know the loft of the putter, the actual angle the ball is launched (could be different from putter loft by hitting up or down), the speed of the ball at launch, the ball weight, the ball size, and the Stimpmeter speed of the specific green at that time. Then you put it all together.

14. Is there a straightforward experiment that a novice could perform, that shows the relation between putting and Physics?

Sure. Galileo studied how a ball rolls down an inclined plane and then rolls up the other side a distance slightly shorter, and the difference is accounted for by air resistance on the ball and by friction between the ball and the plane. One thing he learned was that the higher the ball starts, the faster it is going at the bottom. So you can adjust the speed of a rolling ball by adjusting the height of the start of the inclined plane. Rice University explains all this well at its website page about Galileo's Inclined Plane.

In golf, this has two applications.

First, a Stimpmeter is an inclined plane designed to give all balls the same speed. They then use this to compare the green speed of one green to another, so the greenkeeper can try to get them all about the same speed (for fairness), or make them faster for pros or slower for amateurs. How it is used is the greenkeeper (or tournament official) rolls three balls down the ramp and measures how far they roll across a flat stretch of the green, and then goes to the opposite side and does the same. He then averages the six distance measurements and that is the speed (in feet). A 7.5' green is slower than a 9.5' green. Most amateurs greens are between 8 and 9. Pros play on fast greens. Augusta National at the Masters can sometimes get faster than 12. Note that the green speed can go up or down during the day, depending on mowing, grass growth, rain or fog or dew, air temperature, shade, humidity, and wind. On a dry windy day, even though the grass is growing (which tends to slow the green down as the day goes along), the dry wind and sun draws the ,oisture out of the grass by evaporation and a dry green is a lot faster than a moist green. So, you can make your own Stimpmeter and test greens and compare one green to another and test a single green at different times of day. You can sometimes get a Stimpmeter from a golf course superintendent (borrow one) or get one for real. Take a look at my Greenkeeping page under "Condition" and then look at the Stimpmeter websites. You can also look at my Putting Aids page, at my Homemade Aids, where I describe how a Stimpmeter is constructed and used.

Second, you can explore how a golf ball drops in a hole. A hole is a cylinder shape. You can start instead with a trench shape. Push two tables close together but leave a 4.25" gap. This is the same as a trench that is as wide across as a golf cup at its widest. Use an inclined plane to launch a golf ball and see how high you have to lift the end of the plane to get the ball to skip over the hole. You will have to figure out a design for the ramp that is long enough and that does not make the ball bounce or pop at the bottom, but you can probably figure it out. You can use physics formulas to figure out how fast the ball has to be going, and then compare that speed to the height of the inclined plane to get an idea of how lifting the plane higher gives the ball more energy. The force you use to lift the ball against the force of gravity is basically stored or put into the ball at the top of the ramp. HOW MUCH you put in determines the speed of the ball coming off the ramp, so the speed at which the ball starts jumping the trench should match a particular height for your ramp. You might want to try this in the opposite direction: calculate the jump-over speed, and then try to calculate how high to lift your ramp. (Your teacher will have to help with this.) Then try it and see how closely the calculations pan out with experimiental rolls at that height.

15. Can the physics of putting be compared to the physics of another movement of an object?

Absolutely. Putting and billiards are often talked about together. Hitting a golf driver shot and hitting a baseball are also talked about together. Same physics applied to different cases.

16. How does a full swing relate to Physics?

Same physics. Energy transfer from club motion to ball motion.

17. Is there any specific equations that describe the force at impact?

Sure. The formula comes from the "law" of the Conservation of Momentum and Energy. That is, so far as we know, nature does not do anything funny with energy and everything can always be accounted for if we look hard enough to what happens. Momentum is mass times velocity (speed in a given direction). A golf swing is really curved motion, but the physics is easier if you pretend it is straight-line motion, more like hitting a cue ball with a cue stick. The "law" is that the energy and momentum before impact is THE SAME after impact, since nothing funny happens to the energy. So, you already know the mass of the clubhead (ignore the rest of the club, although in very precise physics you don't) and ball. And you know the ball speed before impact (0). So the only htree numbers not fixed at the start are the clubhead speed before, the clubhead speed after, and the ball speed after (speed really meaning velocity). At golf stores, they often have a machine that monitors the clubhead as it passes over a mat to hit the ball, and this measures the clubhead speed before impact, so that gives you one of the three numbers. It's too hard to measure clubhead speed just after impact without some very expensive high-speed cameras, so look instead at figuring out ball speed after impact. You can figure out average speed from how FAR the ball goes as it flies, by counting the seconds of flight and then measuring to where the ball hit the ground, and divide the feet by the seconds. This is the average ball speed. the ball speed just after impact is the top speed, and the final speed can be thought of as zero (although that's not exactly precisely true). So the top average speed is Top - Final / 2. And thus the top speed is 2 times the average speed. So you have the second number of the three, ball speed just after impact. There's only one left -- clubhead speed after impact, and you can figure this out mathematically from the law of conservation of energy and momentum, since an equation made up of some known numbers and only one unknown variable solves itself. The Law formula is Energy before = Energy after. The energy before = club energy + ball energy. Club energy = club mass times club speed. The energy after = club energy + ball energy. Club energy after = club speed times mass. Ball energy after = ball speed times mass. Putting all this together, Club speed before * club mass + ball speed before * ball mass = Club speed after * club mass + ball speed after * ball mass. That's four numbers and you already have three: 100 mph * 200 grams + 0 * 45 grams = X mph after * 200 grams + top speed of ball * 45 grams. The top speed of the ball is 2 * (distance/seconds of flight). You have to use all feet or all miles but don't mix the units. If club speed is mph, the distance has to be measured in feet or yards and then converted to miles, and then miles per second converted to miles per hour (multiply by the number of seconds in an hour, 3,600). That will give you all the numbers. Then if you add lead tape to your driver, AND this does not slow down your swing speed, the ball will go farther! And if you swing faster, the ball will go farther. And if you use a lighter ball, the ball will go farther!

18. If so, is there any way to test them?

The USGA requires that all club manufacturewrs send their new models to USGA headquarters in Far Hills, NJ, where scientists test the clubs with physics experiments. They want to make sure that clubs don't make the game too easy to play. Golf courses are very expensive to build and operate, so it's not too cool if a company somewhere makes a bazooka driver that allows any hack golfer to drive the ball onto every green. What kind of game would that be? So, both balls and clubs are tested to make sure they are within the speed limits set by the US Golf Association to keep the game fair and fun. Balls that fly too far or clubs that hit too far are made all the time, but the USGA rules them "illegal" so they can't be used for "real" golf under the Rules. Even so, amateur golfers use some of this illegal equipment anyway, because it is more fun for them, and that's fine. You might not want to play against someone using these clubs or balls, though, since they might be able to use them effectively. Maybe not.

19. Breaking down the full swing, how would you describe it?

The full swing is a turning of the body back and through to give the clubhead maximum speed through impact with the ball. Since the body is jointed, it is not a simple turn, but a complex series of coordinated movements of feet, legs, hips, torso, shoulders, arms, and hands (and head) in a arcing of the clubhead back and then down and through impact and into the follow-through and up and back around behind the body. If you take a straight stick like a broom handle and swing it, you want it to make a swoosh sound with the fastest sharpest part of the swoosh right at the bottom of the swing. Different golfers believe in different ways to put all the pieces together. The physics of how all this fits together is described in Theodore Jorgenson's book, The Physics of Golf.

20. If one knows the Physics of golf, is he or she more likely to perform better in the sport?

Yes. When something goes wrong, like a fat shot or a slice, you understand how it happened so you can adjust your technique so it won't happen again (hopefully) and so that you can use techniques that work as well as you can do.

Cheers!

Geoff Mangum
The PuttingZone
http://puttingzone.com
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