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Aerodynamics in Ball Sports
Charlton Lu
Mathematics of the Universe – Math 89S
Professor Bray
1 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:
𝑔(sec(𝜃))2 2
𝑦 = tan(θ) x −
𝑥
2𝑣𝑜2
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.
Figure 1
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).
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 𝐷𝑝
~ 𝜌𝜋𝑎2 𝑈 2 where 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
kinematic viscosity is defined as
𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
𝑣=
𝜇
𝜌
=
𝑈
𝑎𝑣
where 𝑣, the
. 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
𝐴 of 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 𝑠̂ is the direction vector of the sphere’s initial velocity, and 𝑥̇ is 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.
Figure 2
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.
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 𝐶𝐿 is the lift coefficient, 𝐴 is the cross sectional area of the ball, and (𝜔
̂
× 𝑣̂) 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).
Figure 4
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).
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
Acheson, D.J. Elementary Fluid dynamics. Oxford: Oxford University Press, 1990.
Anderson, J.D. "Ludwig Prandtl’s boundary layer." Physics Today (2005): 42-45.
Batchelor, G.K. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press,
1967.
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.
Mehta, Rabindra D. Aerodynamics of Sports Balls. Moffet Field, 1985.
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_HesserKnoll/max_hesserknoll/Slide3.htm
3. http://www.thecompletepitcher.com/different_baseball_pitches.htm