# Flight Trajectory of a Golf Ball

Dimples vs. No Dimples

## By: Kevin Maxson

Dimples. The most recognizable characteristic of a golf ball that separates it from all others. Dimples are taken for granted in the sporting world. They are a miraculous invention that enables golf balls to travel much farther, more accurately, and with a stable trajectory. Yet, the reasons why dimples are effective are unknown for most people. The forces acting upon a golf ball while in flight will be examined in this page, including the effects that dimples have on these forces. These concepts culminate in a computational model that takes these forces into effect to demonstrate the flight trajectory of a golf ball, both with and without dimples.

# Dimples vs. No Dimples

## Turbulent vs. Laminar Flow and its Effect on Drag

The main determinant in making a golf ball is for it to travel farther. The easiest and most efficient way to reach this goal is to reduce the drag force of the golf ball as it is in flight. A common misconception is that a smooth ball will provide the least amount of drag. Smooth balls may be thought to slip though the air easier, with air flowing smoothly around it. However, dimples reduce drag better that smooth balls because it creates turbulence.

Dimples create turbulent flow in addition to the laminar flow created by smooth golf balls. For streamlined bodies, such as race cars or plane wings, laminar flow reduces drag better. In these cases the air around the object slides to the side and back with little difficulty. However, with blunt objects, the air cannot slide back into place as easily. Blunt objects create a large wake, or separated flow, behind it as it moves. The turbulent flow that dimples create allows the air to stay closer to the surface of the ball for longer, thus reducing the separation behind the golf ball. This wake behind the golf ball has low pressure, thus causing deceleration of the golf ball. A smaller wake means less energy needed to pull the stagnant air pocket behind the ball, thus reducing drag. Dimpled golf balls create about half as much drag as smooth golf balls.

## Effect on Lift

Besides reducing drag, dimples also drastically effect the lift of a golf ball. Dimples help grab the air while spinning, amplifying the Magnus Force on the ball. The rough ridges create more airflow around the golf ball, increasing the pressure difference. Dimples effectively create about 2/3 more lift than smooth golf balls.

# Forces Acting Upon the Golf Ball

There are countless numbers of forces that act upon a golf ball through all stages of flight. I will focus on the 4 main forces that affect total distance and are not affected by the environment. These forces are gravity, drag, lift, and buoyancy.

## Gravity

Gravity is the most basic force that acts upon a golf ball. Even in the most simplified golf flight simulators, gravity is taken into account. For our purposes, we will not take altitude into account when determining the force of gravity.

Force(gravity) = m*g

m = mass g = -9.8 m/s^2

Gravity is negative in magnitude as it is a force in the downward y direction. As the mass of golf balls is regulated, this force will be constant in the computational model.

## Magnus/Lift

From bending shots in soccer, to topspin forehands in tennis, to curveballs in baseball, spin drastically affects the flight pattern. This spin creates a pressure gradient around the ball. The ball is accelerated towards the area of low pressure around the ball. For our purposes, the Magnus Force or Magnus Effect is the force exerted by a rapidly spinning object moving through air. Its direction is perpendicular to the velocity of the object. As sidespin does not affect distance a golf ball travels, backspin is the only relevant spin for our model. Consequently, the Magnus force creates lift perpendicular to the direction of motion.

The largest effect that the Magnus Force has on a golf ball is on its height. High spin creates more lift and a higher trajectory. Ideally, there is much higher spin on short shots, as high spin decreases stopping distance by creating a steeper landing angle and reducing rollout. However, excess backspin causes ballooning and loss of distance. Low spin is ideal on long shots, as it promotes a penetrating trajectory, increases rollout, and reduces the effect on wind.

Force(Magnus/Lift) = (1/2)CρAv^2(r*w/v)

where:

C = coefficient of lift

ρ = density of air A = cross sectional area

v = velocity r = radius of golf ball

w = angular velocity/backspin

## Drag

As explained earlier, drag or air resistance is the force put on the golf ball due to its interaction with air molecules. The air resistance can be calculated using the equation shown below.

Force(drag) = (1/2)CρAV^2

where:

C = drag coefficient (see graph below

ρ = air density A = area

V = velocity

Drag acts opposite of the direction of motion and is the most limiting factor of distance. At the velocity of our model, the drag coefficient of a smooth ball is around .49. For dimpled, the drag coefficient increases at lower Re values/lower velocity but range from .26 to .47.

## Buoyancy

Buoyancy is a very minor force on a golf ball. It is a force of lift in the positive y direction. It is equal to the weight of air that it displaces.

Force(buoyancy) = volume*ρ(air)

# Computational Models

These two computational models below take use the four forces above to calculate the total carry distance, maximum height, and hang time of a golf ball trajectory. The user can input initial speed, launch angle, and backspin at the top of the computational models. These inputs are in the common units of a golf simulator.

The following assumptions were made to simplify the dimpled ball model.

1. No wind

2. No change in air conditions due to altitude or humidity

3. No changes in Barometric pressure

4. No sidespin

Here are some possible test values for the models, based on an average 10 handicap male.

Driver: 152 mph, 14.5°, 3300 rpm

6 iron: 118 mph, 17.0°, 6100 rpm

9-iron: 98 mph, 24.2°, 8400 rpm

# Model With Dimples

# Model Without Dimples

# References

http://www.golf-simulators.com/physics.htm

http://blog.trackmangolf.com/pga-tour-averages-sta...

http://home2.fvcc.edu/~dhicketh/DiffEqns/Spring11p...

http://www.ima.umn.edu/~arnold//papers/golf-flight...

http://math.ucr.edu/home/baez/physics/General/golf...