**Introduction**

According to the PGA there are 27 million golfers in the United States (citation). By understanding the science behind the game, golfers are more likely to improve their scores because they can better understand the errors that they are making.

**From Tee to Fairway: The Physics Behind Golf (Projectile Motion)**

Projectile motion is the motion of an object that is imparted with an initial velocity (such as hitting a golf ball with a golf club) that moves in a parabolic trajectory that is caused by the effect of gravity on the object. Initial velocity can be broken into horizontal and vertical components as in Equation (1).

Where the subscript 0 denotes initial, x and y are the directions and **i **and **j** are the vector directions (**i** is in the x-direction and **j** is in the y-direction). The initial horizontal and vertical velocities can be determined if the angle the ball is launched from the horizontal,α, is known, given by Equations (2) and (3).

Acceleration due to gravity pulls the golf ball down during flight and as time goes on actually forces the vertical velocity in the downward direction, the components of the golf ball velocity at any given time, t, during its flight are given by Equations (4) and (5).

Where g is the acceleration due to gravity, which is a constant assumed to be -9.8 m/s^{2 }(or 32 ft/s^{2}), and the time t is in seconds. The magnitude of the velocity is given by Pythagorean’s theorem.

A diagram of these velocity vectors is given in Figure 1.

The average golfer drives the golf ball with an initial velocity of over 100 miles per hour! (Zumerchik, 2002). This means that the golf ball, if struck at an angle of 12°, will be initially travelling at 87 mph in the horizontal direction and 13 mph in the vertical direction!

Typically the projectile motion equations used to calculate range and height of a golf ball’s flight do not account for drag, and are therefore only estimates as to the true maximum height and range that the golf ball will travel.

Drag force (or air resistance) is the force that acts opposite to an object that is moving through it. When a golf ball is hit, the air molecules flow past the golf ball as the golf ball flies through the air, thus creating a retarding force on the forward motion of the ball known as drag.

In 1949, Davies conducted experiments to determine the magnitude of drag and lift forces that occur on a golf ball by dropping rotating golf balls into a wind tunnel. Davies found that “drag increased nearly linearly from about 0.06 lb for no spin to about 0.1 lb at 8000 rpm” and that “lift varied with the rotation speed .” (Davies, 1949)

In 1959, Williams used the previous findings of Davies and conducted an analysis on golf ball carry as a function of velocity. Williams showed that the drag force varied linearly with the velocity. The drag force on a golf ball can be calculated by Equation (7).

Where C_{D} is the drag coefficient, A is the cross-sectional area of the ball, ρ is the air density, and v is the ball velocity (in ft/s).

Williams showed that the drag coefficient can roughly be estimated as 46/v, showing that at higher speeds the drag coefficient drops significantly. Williams found that the drag force on the golf ball varied linearly with the speed, he found D to be as given in Equation (8).

Where D is in pounds, and 0.000783 represents the constant c, to be discussed. One note is that Williams’ (1959) calculations used a British ball with a diameter of 1.62 inches compared to the American ball of 1.68 inches. The easiest calculation is to consider the case of a nonspinning golf ball (no lift) with linear air resistance in calculations of range and height. The reader is directed to the paper by Erlichson for a more in depth analysis of range calculations that incorporate lift forces into the derivations.

Erlichson (1983) gives the equations for range and height as in Equations (9) and (10).

Where c is 0.000783 lb/(ft/s) and m is in [lb] and g is 32 ft/s^{2} and t is time, initial velocities are given in ft/s.

Using these equations a golfer could estimate how long they will hit the ball and where it will end up. Variables such as wind and weather can affect these numbers in reality so they are best used for only estimations on distance.

Several studies have been conducted on the optimum launch angle. Scottish physicist Alastair Cochran calculated that the optimum launch angle of 20° achieves the longest carries. (Cochran, 1990) One needs to take this finding with a grain of salt, however, because the ball will land at a high angle and will have less bounce and roll, unless the grass is wet.

Erlichson (1983) found that the optimum launch angle was around 12-13°, only slightly more lofted than the drivers available on the market, and with the additional loft generated from shaft flex. Zumerchik concludes that anywhere from 12°-20° will give the ball maximum range with only a few yards difference between the different angles. (Zumerchik, 2002)

**Golf Clubs: Loft and Grooves**

Several forces act on the golf club, such as torque (exerted by the golfer on the club), centrifugal acceleration, and gravity. Typically golfers can average a whole 4-5 hP of power generated from their golf swings. (Wesson, 2009)

At the top of the backswing, the club head coils because the shaft is flexible and the center of mass is in the clubhead. As the swing moves forward, the shaft of the club coils, unloads, and recoils as the club head attempts to catch up to the wrists during the swing. At impact between the clubhead and the ball there is a final forward oscillation that creates a snapping effect that increases the velocity of the club head through the ball, and by extension also increases the initial velocity imparted to the golf ball. It has been found that a “forward shaft flex of about 3.3 degrees can add 8.7 percent to the velocity of the club head.” (Jorgensen, 1994). During the course of the swing the club oscillates 1.5 times. (Zumerchik, 1997)

Werner et al. (2000) conducted a full suite of simulations and experiments on optimizing club designs. Their findings showed that there is an optimum combination of loft angle and center of gravity location, and that an extra-large face provides advantages (Werner et al, 2000) But does the addition of more surface area on the club head create an increased drag on the club head, diminishing initial velocity?

It’s been found that air drag on the shaft and clubhead results in an energy loss of 10% of the energy of the club at impact with the ball. This results in a reduction of roughly 15 yards in the range of the ball (Wesson, 2009)

When the club head strikes the ball, the grooves on the club head increase the amount of friction at the ball-club head interface, allowing for the ball to have backspin, which increases the lift force. Without friction, the ball would merely glide up and off the club face.

When the club head comes into contact with the golf ball, some of the energy is lost and is not entirely transferred into velocity of the golf ball. The ratio of the velocity transfer between the club face and golf ball surface is given by the coefficient of restitution.

If a ball hits a surface at a right angle with a speed v and leaves the surface with a speed v’ then the coefficient of restitution is defined by

When bounced off a hard surface, a typical golf ball has a coefficient of restitution of around 0.7. This means that if the golf ball were dropped from 100 feet above the ground, it would rebound and bounce back 70 feet. The coefficient of restitution is about 1.46 for a driver, 1.3 for a 5 iron, and 1.12 for a 9 iron. (Zumerchik, 1997).

**Why all the dimples: The Fluid Mechanics behind a golf ball in flight**

The aerodynamic forces that act on a golf ball in flight are shown in Figure 1.

Typically golf balls have between 330 and 500 dimples on their Surlyn covers. Some dimples are round while most golf ball manufacturers have started making their dimples in a variety of hexagonal shapes. As stated by Euler’s principle (Bird et al, 2007) separation of the boundary layer (the layer of air molecules next to the surface of the golf ball) is likely to occur in regions where the pressure increases in the direction of the flow. The following youtube video gives a good look at a simulation of the boundary layer. Click here to watch.

The reason for the large number of dimples on a golf ball is to ensure that the boundary layer does not separate until the back part of the ball. A smooth sphere will generate a large wake behind the ball, as there is much lower pressure behind the ball than in front. The air will move to rush into the area of low pressure, exerting a pressure drag force on the ball. However, as the turbulent air swirls around the golf ball, the dimples capture some of the swirls and keep them close to the surface of the golf ball. Dimples force a turbulent trip at the surface, ensuring a later separation of the boundary layer which decreases the amount of drag force on the ball. Dimples on the golf ball do not reduce the drag at the front of the ball, because the cross-sectional area of the ball is always the same. However, they do decrease the size of the low pressure wake behind the ball, lowering the overall drag force, and improving flight. An example of this is shown in Figure 2, a simulation run in Fluent between a sphere and a dimpled golf ball.

As shown in both cases, the pressure contacting the front face of both balls is large. However, the magnitude of the pressure difference (front side versus back side of the ball) in the sphere case is much greater than in the golf ball case. The large amount of high pressure (red and yellow) pushing against the flight of the ball slows it down much faster than in the golf ball case, where there is less pressure behind the ball. In addition to this pressure calculation it is of interest to look at the shear stress distribution on the face of both the sphere and modeled golf ball, as shown in Figure 3.

As shown in Figure 3, the shear stress on the surface of the sphere (red/orange) covers a greater surface area, showing where the boundary layer separates from the sphere. The dimples on the golf ball keep the turbulent flow boundary layer close to the golf ball wall and greatly reduce the shear stresses over the golf ball’s surface area. This means that the golf ball is slowed less. The boundary layer separation in laminar and turbulent flow is shown in Figure 4, a COMSOL simulation of a sphere in flow.

As shown in Figure 4, in the laminar case with a low Reynolds number (Re=0.1) (top) has no boundary layer separation as the streamlines hug the outside of the ball. This situation is unreleastic for a golf ball because of the high velocities that the golf ball travels at (~200 ft/s) once it is hit. (Zumerchik, 2002)

The second case (middle) shows a smooth ball at a high velocity. There is a large pressure difference between the front and back side of the ball, and the boundary layer separates much earlier than the dimpled sphere case (bottom). The dimples allow for the boundary layer to hug the ball and boundary layer separation does not occur until the very tail end of the ball. The smaller the blue zone of low pressure behind the ball, the farther the golf ball will fly.

Similar to an air foil, the dimples on a golf ball also allow for a lift force to be exerted on the ball. Backspin, as generated from the loft of the clubface that strikes the ball, deforms the airflow around the ball and creates a lift force due to the Magnus effect. The Magnus effect is a phenomenon where a spinning object flying in a fluid creates a whirlpool around itself and experiences a force (in this case, lift) perpendicular to the line of motion. Because the top of the ball is spinning with the direction of the air, the air on top of the ball moves more rapidly than the air at the bottom of the ball. The air at the bottom of the ball moves against the wind, and this shifts the pressure behind the ball downward, in the direction of the backspin. Because of the differing velocities on top and bottom of the ball, there is a resultant force upward known as lift.

The magnus effect can have a large impact in golf. If two golf balls are hit with the same velocity, a ball hit with backspin will stay in the air 2 or 3 seconds longer and may travel 18 to 30 meters farther (Zumerchik, 1997).

Despite the benefits of backspin, the use of too much spin can be a problem because some of the momentum is imparted into the high spin of the golf ball. As a ball travels faster, it needs less backspin to generate lift. (Zumerchik, 2002).

**Conclusions**

Golf equipment has evolved over the centuries to increase the distance a ball can travel. By understanding the forces acting on the golf ball, a golfer can maximize his/her performance

**By: Trevor Stoddard, University of Utah**

**References:**

- Benson, Tom(2010) Drag of a Sphere. National Aeronautics and Space Administration, Date Accessed: 8/10/2012 < http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html>
- Bird, R. Byron, W.E. Stewart and E.N. Lightfoot,
*Transport Phenomena*, 2^{nd}edition, Wiley & Sons, New York, 2007. - Cochran, A. (ed.) (1990) Science and Golf. New York: Chapman and Hall
- Cochran, A. (ed.) (1992) Science and Golf II. New York: Chapman and Hall
- Davies, J. (1949) “The Aerodynamics of Golf Balls.” Journal of Applied Physics 20: 821-828
- Erlichson, H. (1983) “Maximum Projectile Range with Drag and Lift, with Particular Application to Golf.” American Journal of Physics 51: 357-362.
- Jorgensen, T. (1994)
*The Physics of Golf.*New York: American Institute of Physics - McDonald, W. (1991) “The Physics of the Drive in Golf.” American Journal of Physics 59: 213-218
- Werner, F. and R. Greig. (2000)
*How Golf Clubs Work and How to Optimize Their Designs.*Jackson Hole, WY: Origin Inc. - Wesson, J. (2009)
*Science of Golf*. New York, Oxford University Press - Williams, D. (1959) “Drag Forces on a Golf Ball in Flight and Its Practical Significance.” Quarterly Journal of Mechanical Applications of Mathematics XII 3: 387-393
- Zumerchik. J. (ed.) (1997) Encyclopedia of Sports Science
- Zumerchik J. (2002) Newton on the tee- a good walk through the science of Golf
- History of the golf ball <http://www.golfeurope.com/almanac/history/golf_ball.htm> last accessed 8/10/12

**For more information:**

- DeNevers, Noel,
*Fluid Mechanics for Chemical Engineers*, McGraw-Hill, New York, 2005. - Libii, Josue Njock (2007) “Dimples and Drag: Experimental demonstration of the aerodynamics of golf balls.” American Journal of Physics,
**75**, 764. - Millne, R. and J. Davies (1992) “The Role of the Shaft in the Golf Swing.”
*Journal of Biomechanics*129: 975-983 - Zander, J. and A. Chou (February 1999) “Max out your ball: Increasing your launch angle and decreasing your spin rate will help you hit farther.” Golf Digest 50: 76-80

*,*