Aerodynamics in Ball Sports

Charlton LuMathematics of the Universe – Math 89S

Professor Bray1 November 2016

Introduction:

Simple physics will tell us that if we throw a ball in the air, its trajectory will be a perfect

parabola. However, if we were to trace the trajectory of a baseball, tennis ball, or golf ball

travelling through the air, we would find a variety of shapes, some that differ drastically from a

parabola. In fact, this variety of shapes is an integral part of the game: anyone who watches

sports will witness curveball pitches in baseball, topspin shots in tennis, and shot-shaping in golf.

None of the sports would be the same without the nuance that aerodynamic properties add to

each game. The mechanism that causes the curvature requires a more comprehensive model than

simple projectile motion. A basic application of Newton’s laws ignores the effects of air

resistance among other aerodynamic effects. However, the geometry and dynamics of each sport

are engineered to take advantage of aerodynamic properties, and their effects are far from

negligible. This paper aims to explain the physics behind those properties while providing a

mathematical model for each projectile’s flight.

Projectile motion in a vacuum:

If we first ignore the effects of air resistance and friction, we can model the motion of a

ball travelling through the air with a simple implementation of Newton’s second law. With the

sole force of gravity acting on an object, we can find the differential equation:

In this equation, m is mass, x double dot is the second derivative of position (acceleration), and g

is the gravitational acceleration. When the equation is solved, it yields the equation:

y=tan (θ ) x−g (sec (θ ))2

2vo2 x2

which, when given initial values of velocity (vo) and angle of release (θ), graphs a parabola. This

proves that the projectile path of an object under the sole influence of gravity is parabolic (Bush).

However, we know from personal experience that baseballs, tennis balls, and golf balls only

roughly follow a parabolic path.

For example, Figure 1 shows a drastic decrease in distance when a smooth sphere travels through

air instead of a vacuum. In addition, the shape of the sphere’s trajectory when travelling through

air is not perfectly symmetrical like a parabola but rather has a rightward bias. The discrepancy

can be accounted for in part by adjusting for air resistance.

Projectile motion when accounting for air resistance:

When an sphere travels at speed U through a fluid, it experiences a resistive force with

two components: viscous drag and pressure drag. Viscous drag is a resistive force caused by a

fluid’s resistance to flow. An object in movement forces the fluid to displace but encounters a

reactionary friction force that decreases the velocity of the object (Institute of Hydrodynamics).

The force exerted by the fluid on a sphere is proportional to Dv μUπa where μ is viscocity, U is

the velocity of the sphere, and a is the radius of the sphere (Bush).

Figure 1

On the other hand, pressure drag is caused by a difference in pressure between the front

and the back of the sphere. When the fluid is displaced by the sphere, there is an area of high

pressure in front of the sphere due to the compression of the fluid when it comes into contact

with the sphere. Behind the sphere, the fluid stream has a lower pressure because the flow of the

fluid separates from the sphere, creating an area with a lower concentration of fluid particles.

The higher pressure exerts a greater force in the front of the sphere than the force caused by the

lower pressure in the back. The net force acts in the opposite direction of the sphere’s velocity,

and is proportional to D p ρπ a2U 2where ρ is fluid density.

From the two types of drag forces, we can find the Reynolds number—a dimensionless

ratio between pressure drag and viscous drag—given by the formula Re=Uav wherev, the

kinematic viscosity is defined as viscos itydensity

v=μρ . The Reynolds number tells us the relative

effects of pressure drag and viscous drag. A high Reynolds number indicates that pressure drag

accounts for most of the overall drag force (Bush).

Using the Reynolds number and the geometry of the sphere, we can now solve for the

drag coefficient, given by the formula:

The drag coefficient is an experimentally derived formula that models the effect of air resistance.

Rearranging, we can find the drag force.

We can now see that the drag force is proportional to the density of air, frontal area Aof the

object, the square of the velocity, and a function of the Reynolds Number. With a more

comprehensive approximation of the sphere’s trajectory, we can now rewrite the original

differential equation by plugging in for the sphere’s frontal area:

where s is the direction vector of the sphere’s initial velocity, and xis the first derivative of

position (velocity). Solving and graphing this differential equation provides a more accurate

model of a sphere’s trajectory. Inputting realistic values for density and the Reynolds Number,

we can find a trajectory similar to green graph found in Figure 1 (Bush).

Magnus Effect

Though we now have a much better approximation of a sphere’s trajectory when

travelling through a fluid, Figure 1 shows that a golf ball travelling through the same fluid with

the same starting velocity will fly much farther than a smooth ball. In addition, we cannot yet

explain the drastic curvature of a baseball, tennis ball, or golf ball’s flight through air resistance

alone; there must be a missing force.

The missing force comes in the form of the Magnus Effect. The Magnus Effect is the

tendency of a spinning object to curve away from its expected flight path. It comes from the

Magnus Force, which arises due to the spin of the sphere and acts in a direction orthogonal to

both the angular velocity and the velocity vectors. In order to find the direction of the Magnus

Force, we must use the right hand rule. If we point our fingers in the direction of the angular

velocity vector and curl them towards the velocity vector, the direction of our thumb will yield

the direction of the Magnus Force. In more mathematical terms, we are finding the direction

vector of the cross product between angular velocity and velocity (Briggs).

Real examples can make the abstract physics more clear. A tennis ball hit with topspin

will have an angular velocity vector parallel to the ground and towards the left if we are looking

at the back of the ball. The velocity vector will vary but always have a positive x component of

velocity, as the ball travelling through air will always travel forwards. By applying the right hand

rule, we find that the direction of the Magnus Force varies but will always have a negative y

component—that is, the Magnus Force will pull the tennis ball down, reducing the distance

travelled. On the contrary, a ball hit with backspin (underspin) will have the opposite effects. The

angular velocity vector will be towards the right, leading the Magnus Force to have a positive y

component that increases the distance travelled by the ball.

Professional tennis players such as Roger Federer and Rafael Nadal can take advantage

of the Magnus Effect. By using heavy topspin, they can hit a ball with a very high velocity

without worrying about the ball travelling too far. They can also use backspin to keep the ball’s

trajectory low and flat, making it difficult for their opponent to return their shot.

Figure 2

This Magnus Force is caused by a difference in pressure. When a sphere spins while

travelling through a fluid, it imparts a force on the fluid tangential to its spin direction. In the

example of a tennis ball with backspin, the top of the ball will spin in the same direction as the

air stream, accelerating the air stream downwards and creating a low pressure area. The bottom

of the ball will spin in the opposite direction of the incoming stream, reducing the stream speed

and creating a high pressure area. The gradient in pressure will lead to an upward Magnus Force

that pulls the ball up (Figure 3). The Magnus Force can also be understood through Newton’s

Third Law. When the ball pushes the air stream down, there must be an equal force that pushes

the ball up, causing the ball to lift (Dyke).

Figure 3

The effect is even more apparent in golf, where the ball spins backwards at thousands of

revolutions per minute when hit with a club, and the geometry of the golf ball is optimized to

maximize the Magnus Effect. On the surface of a golf ball, there are between 300 and 500

dimples. These small indentations maximize the force that the ball imparts on the fluid—they

can increase and decrease the speed of the air stream more significantly due to the higher force

of friction and normal force between the ball and the fluid. As a result, there is a larger pressure

gradient between the bottom of the ball and the top of the ball, thereby increasing the lift force.

In addition, the dimples catch the air stream and carry it with the curvature of the ball. As a

result, the pressure behind the golf ball is higher than that of a perfectly round sphere. This

decreases the pressure drag, allowing the golf ball to travel much further (Davies). Looking back

to Figure 1, we see that a golf ball with the same initial velocity as a sphere can travel

significantly further if it has backspin because of the Magnus Effect. We can approximate the

Magnus Force with the formula:

where CLis the lift coefficient, Ais the cross sectional area of the ball, and (ω× v )is the cross

product between angular velocity and velocity. The lift coefficient is a value that depends upon

the geometry of the ball and the ratio of rotational speed to translational speed. Using the force

equation for the Magnus Effect, we can come up with a much more accurate model of a ball

travelling through the air (Bush).

Sports Applications of the Magnus Effect

Our analysis of the Magnus Effect thus far has assumed that the projectile’s direction of

spin is either towards or away from its direction of motion (backspin or topspin). Often in sports,

athletes utilize a greater range of spin directions. For example, a pitcher in baseball will draw

upon his repertoire of pitches (Figure 4) to confuse the batter and hopefully cause the batter to

miss. He can throw a curveball, which uses topspin to suddenly pull the ball down before home

plate. He can also throw pitches with sidespin so that the Magnus Effect acts in an unexpected

direction. The slider pitch in baseball, for instance, is thrown with heavy sidespin, resulting in a

trajectory that curves away from a right hand hitter if thrown by a right hand pitcher. By

switching between different directions of spin, the pitcher can drastically change the trajectory of

the pitch and make it very difficult for the batter to hit the ball (Richmond).

Similarly, experienced golfers can strategically impart spin on the golf ball to shape the

ball flight. A ball struck with a closed clubface by a right handed golfer will have two effects.

First, the ball will be experience the impact of the club on the right side first, causing a counter

clockwise spin. Second, closing the clubface coincides with a decrease in backspin because of

the decrease in loft—the club’s angle relative to the ground. The result is a draw: a lower ball

flight that curves from right to left. The opposite is a fade: a ball struck with an open clubface

will have a clockwise spin and more backspin, leading the ball flight to be higher and curve from

left to right (Davies).

Figure 4

Reverse Magnus Effect

After our discussion of the Magnus Effect, we can expect a beach ball with backspin to

have an upwards Magnus Force. However, the effect is actually the exact opposite; a beach ball

or any other very smooth ball travelling with spin actually moves in the opposite direction of that

predicted by the Magnus Effect. The effect is due to the geometry of the smooth ball. If we have

a smooth ball moving with backspin, the top of the ball will spin in the same direction as the air

stream. Unlike the case of a rougher ball, the air stream will not follow the curvature of the ball

but rather flow right past the ball—the friction on the surface of the smooth ball is not enough to

push the airflow downwards. On the other hand, the airflow at the bottom will come into contact

with the surface of the ball and become turbulent—airflow characterized by chaotic changes in

pressure and flow velocity. This turbulent airflow is pushed with the curvature of the smooth

ball, as the chaotic movement of air particles will lead to more frequent and higher energy

transfer interactions between the air and the surface of the ball, allowing a combination of the

normal force and the friction force to push the airflow upwards. In the case of a rough ball with

backspin, we have the same turbulent airflow at the bottom of the ball, but the difference lies in

the net effect. For a rough ball, the turbulent airflow at the bottom of the ball is outweighed by

upward force caused by airflow at the top of the ball. The effect on a smooth ball is the opposite:

a beach ball with backspin will have a net downward force and curve downwards (Bush).

Conclusion:

By accounting for air resistance and the Magnus Effect, we can understand the

unabridged version of sports physics. The curvature in trajectory of tennis balls, golf balls, and

baseballs all are subject to the same laws, and the tools we used to analyze their flight can be

extended to other projectiles—from soccer balls to bullets, each projectile can be modeled by a

similar force analysis.

Works Cited

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Briggs, Lyman J. Effect of Speed and Spin on Lateral Deflection (curve) of Baseball; and the Magnus Effect for Smooth Spheres. Washington D.C., 26 March 1959 .

Bush, J.W.M. "The aerodynamics of the beautiful game." Annual Review of Fluid Dynamics (2013): 151-189.

Davies, J.M. "The aerodynamics of golf balls." J. Appl. Physics (1949): 821-828.

H. M. BARKLA, L. J. AUCHTERLONIET. The Magnus or Robins effect on rotating spher. St. Andrews, 16 November 1970.

Institute of Hydrodynamics. MAGNUS AND DRAG FORCES ACTING ON GOLF BALL. Prague, 24 Oct 2007.

Matthews, John H. Module for Projectile Motion. Fullerton, 2004.

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Raymond Cho, Samual Leutheusser. Modeling The trajectory and measuring the Magnus coefficient and force of a spinning ping pong ball. Vancouver, 30 4 2013.

Ricardo, Julian. "Modeling the Motion of a Volleyball with Spin." Journal of the Advanced Undergraduate Physics Laboratory Investigation 2.1 (2014).

Richmond, Michael. The effect of air on baseball pitches. 25 May 2011. <http://spiff.rit.edu/richmond/baseball/traj_may2011/traj.html>.

Picture bibliography

1. http://aolab.phys.dal.ca/~tomduck/classes/phyc2050/

2. http://ffden-2.phys.uaf.edu/webproj/211_fall_2014/Max_Hesser-Knoll/

max_hesserknoll/Slide3.htm

3. http://www.thecompletepitcher.com/different_baseball_pitches.htm