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Football in Flight:
A study of the math and physics of the trajectory of a kicked
football
Ryan Swenson
April 2, 2013
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Abstract
Three seconds left, down by two, one player will determine the
outcome of the big game; one player will make or break his teams Super
Bowl dream. The kicking game associated with the sport of American
football is a crucially important aspect of this American pastime. Many
factors affect the outcome of a kick, whether it be a last-second field goal,
the game-beginning kickoff, or a fourth down punt. Basic factors affecting
the trajectory of the ball include the launch angle and velocity of the kick.
A much more complicated factor, often ignored by math and physics
classes is the force due to air resistance. Finally, the factor most
memorable to the kicker that affects the outcome of a kick is the wind.
Crosswind, tailwind, or headwind, this factor is the bane of many kickers
glory. In this study we explore the launch angle and initial velocity of a
kick, the force due to air resistance on a football, and the effect that wind
has on the trajectory of a football. Our resulting model approximates the
trajectory of a kicked football, whether it be through a field goal, kickoff,
or punt. Taking into account air resistance, crosswinds, headwinds,
tailwinds, various initial velocities, different kick angles, and any
geographic location, our model can plot the trajectory of any kick.
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CONTENTS Swenson
Contents
1 Introduction 6
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Pro jectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Models 7
2.1 Vertical Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Horizontal Motion . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Drag Forces 9
3.1 Coefficient of Drag . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Air Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Differential Equations of Motion 15
4.1 Eulers Method and Excel . . . . . . . . . . . . . . . . . . . . 15
4.1.1 Field Goals . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.2 Kickoffs . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.3 Punts . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Wind 23
5.1 Crosswinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Headwinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3 Tailwinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Conclusion 30
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CONTENTS Swenson
6.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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LIST OF FIGURES Swenson
List of Figures
1 Spiral Air Resistance . . . . . . . . . . . . . . . . . . . . . . . 13
2 Tumble Air Resistance . . . . . . . . . . . . . . . . . . . . . . 14
3 Max Air Resistance . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Excel Screenshot . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Field Goal Launch Angle . . . . . . . . . . . . . . . . . . . . 18
6 Field Goal Tra jectory . . . . . . . . . . . . . . . . . . . . . . 19
7 Field Goal Low Trajectory . . . . . . . . . . . . . . . . . . . . 19
8 KickoffTra jectory . . . . . . . . . . . . . . . . . . . . . . . . 21
9 Punt Tra jectory . . . . . . . . . . . . . . . . . . . . . . . . . . 22
10 Crosswind of 20fts
Affecting a Field Goal . . . . . . . . . . . . 25
11 10mph Crosswind on a Field Goal . . . . . . . . . . . . . . . 26
12 Effects of a 6.8mph Headwind on a Kickoff . . . . . . . . . . 27
13 Effects of a 13.6mph Headwind on a Kickoff . . . . . . . . . . 28
14 Effects of a 6.8mph Tailwind on a Kickoff . . . . . . . . . . . 30
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Swenson
1 Introduction
1.1 Background
Friday nights, Saturday mornings, Sunday afternoons, Monday
evenings, and the all important Super Bowl. Millions of Americans crave
the hard-hitting, heart-pounding, action-packed lifestyle that is football.
Whether it be watching the home town heroes winning a crosstown high
school rivalry, alma mater bringing home a conference championship, or a
favorite professional team fighting for a Super Bowl ring, football games
are an ever-popular source of camaraderie, excitement, and entertainment.
In addition to being an exciting way to spend a few hours, football is also
a science jackpot. The physics and math involved in the bone-crunching
hits, the mind-bending speeds, or the beautifully-spiraled passes executed
by the players are an endless source of fun for math aficionados in addition
to die hard sports fanatics. The kicking involved in a football game often
includes some of the most exciting, nail biting action in the game. From
kickoffs to field goals, the almost mundane point after touchdown (PAT)
attempts, to the last-ditch punt, kicking is an important aspect of the
football game and brings with it a wide array of mathematical concepts.
1.2 Projectile Motion
Projectile motion is simple for a perfectly spherical ball that is not
affected by air resistance. For every added complication, the mathematics
describing the behavior of the ball becomes more and more sophisticated.
For example, the mathematics of the trajectory of a kicked football is
influenced by many factors: the mass of the ball, the orientation of the
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ball while in flight, air resistance, the spin of the ball, and the force with
which it is kicked. To examine this complicated real-world situation, we
start with the simplest model possible disregarding shape, spin, and mass
of the ball, as well as air resistance. These factors will be added to the
model later.
2 Models
2.1 Vertical Motion
The most basic form of projectile motion describes an object dropped
from a particular height. The equation that models this behavior is
Z= g
2t2 +h0 (1)
whereZ is the vertical position of the object in feet at time t, in
seconds, and h0 is the height (in feet) from which the object is dropped.
The acceleration due to gravity (g) will be measured as 32 fts. In this
simple model, the only force acting on the object is that of gravity. This
model could be expressed in a very similar manner using the metric
system, however because an American football field is measured in yards
we will use the Imperial system.
Adding a velocity at which the object is launched vertically into the air
is the next logical step in complicating and therefore improving the
model.
Z= g
2t2 +v0+h0 (2)
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2.2 Horizontal Motion Swenson
This equation introduces an initial vertical velocity, to account for a
situation where an object is launched upward at some initial velocity v0
from an initial height. This model does not allow any launch angle other
than = 90, and is still limited in its potential applications.
2.2 Horizontal Motion
In order to more accurately represent projectile motion and to model
horizontal movement as well as vertical, we need now two equations, one
each for vertical and horizontal components of motion. These equations
are similar to Equation 2, however they allow for a launch angle other
than = 90.
Motion towards the goal posts
Vy =v0cos() (3)
And motion upwards
Vz =v0sin() gt (4)
These equations use the sine and cosine of the launch angle to
determine how much of the initial velocity is in the yand zdirections,
and therefore the yand zcomponenets of the velocity. [1] For
example, an object launched at a 60angle at a velocity of 75 fts
will have
horizontal and vertical velocity components as follows:
vx= 75cos(60) = 37.5ft
s (5)
and
vy = 75sin(60) 32t= 64.95ft
s
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The vertical velocity component will change as the acceleration due to
gravity acts upon the object, and will decrease the vertical velocity if the
vertical velocity is a positive value, or increase the vertical velocity if the
vertical component is a negative value. The horizontal component will
stay constant throughout the flight of the projectile, having no outside
forces acting upon the object other than gravity. Air resistance is an
outside force that would act upon the objects horizontal velocity, however
in this model air resistance has been negated. As a result of this,
horizontal velocity remains constant throughout the flight of the projectile.
3 Drag Forces
Air resistance, often disregarded in math and physics classes, plays a
crucial role in the flight of a projectile. The projectile is hindered by the
air in all parts of its flight. In this study we will assume that air always
exerts a force on the projectile opposite that of its direction. If the
projectile is going up, the air is pushing it down, but if it is falling, the
force due to air is working to hold the ball in the air (unsuccessfully). A
function to model the drag force on a football is
Fdrag =1
2CADv2 (6)
[2]
3.1 Coefficient of Drag
The first parameter in our equation for air resistance on a football is C
which represents the coefficient of drag on the football due to its shape.
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3.2 Area Swenson
This is a constant value representing the effect that the shape of a
projectile has on the force upon it due to air resistance. We found no
published drag coefficient for a football readily available, but the drag
coefficient of an ellipsoid traveling nose-first is C= 0.1 and the coefficient
for an ellipsoid traveling with its long axis perpendicular to the flight path
is C= 0.6 [5] For a football traveling in a spiral, C= 0.1 will be a close
approximation, but for a football tumbling end-over-end as is typical of a
kickoffor field goal, the average of these two coefficients will be used.
Ctumbling =0.1 + 0.6
2 = 0.35 (7)
3.2 Area
The next parameter in our drag function is A, which represents the
cross-sectional area of a football perpendicular to the flight path. For a
Wilson NFL football, the diameter at the fattest part of the ball is 6.8,
and the cross-sectional area of a football traveling nose-first (as in a
spiraling pass) is a circle with A= r2 = 0.25ft2. The area of a football
traveling with its long axis perpendicular to the flight path is an ellipse
with area A= 0.41ft2. [4] To find the cross-sectional area of a football
traveling end-over-end as in a tumbling kickoff, we will use the average of
these two areas, represented in Equation 8.
A= 0.25ft2
+ 0.41t2
2 = 0.33ft2 (8)
Calculating this average assumes that the ball rotates at a constant
velocity and spends equal time with the long axis perpendicular to and
parallel to the flight path. We will use this new value for A in Equation 6
when the ball is tumbling end-over-end rather than in a spiral.
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3.3 Air Density Swenson
3.3 Air Density
The final parameter in the function for drag due to air is D ,
representing the air density. Air density in any particular location is
dependent on several factors including elevation, temperature, and
humidity. [3] For an average summer day in Helena, MT with a
temperature of 70 F, humidity of 30% and at an elevation of 3875 feet,
the air density coefficient is D= 0.064 slugsft3
. [7] In Denver, CO, at the
Mile High Stadium however, the average temperature is around 78 F
and the afternoon humidity is up around 40%. At an elevation of 5280
feet, the air density coefficient in Denver is D= 0.0741 slugs
ft3 . [6] Although
Denver is at a higher altitude and seems therefore like it would have a
lower air density, the higher humidity plays a big role in the calculations
and in the end creates a higher air density than that of Helena.
3.4 Force Equations
Using our new values for the coefficients C, A, and D (the air density
for Helena, MT will be used throughout this study) we can express the
force on a football in flight due to air resistance.
For a football traveling in a spiral
Fdrag =1
2CADv2 (9)
Fdrag = 12 (0.1)(0.25)(0.064)v2
= 0.0008v2
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3.4 Force Equations Swenson
These three different styles of kicks generate very different forces due to
drag, and have very sizable effects on the trajectory of the football. For
example, Figure 1 illustrates a ball flying in a perfect spiral as is typical
of a well kicked punt after being launched at 50 with a velocity of 85 fts
Figure 1: The trajectory of a football kicked in a perfect spiral
A football kicked end-over-end will have a larger cross-sectional area
perpendicular to the direction of motion and therefor a larger force due to
air resistance, in turn affecting the trajectory substantially more. Figure 2
shows one such trajectory. This ball, like the previous one, was kicked
with an initial velocity of 85 fts
at an angle of 50.
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3.4 Force Equations Swenson
Figure 2: The trajectory of a football kicked end-over-end, as in a kickoffor field goal
Finally, a football kicked in such a way that the balls long axis is
perpendicular to the direction of travel for the duration of the flight will
have an even greater force due to air resistance and consequently will have
a shorter flight. This would be the least desired kick type because of its
high air resistance. Figure 3 demonstrates the trajectory of this type of
kick, and it can be seen that the max range of such a kick is much shorter
than that of a tumbling or spiraling ball.
Figure 3: The trajectory of a football kicked such that its long axis is perpendicular to the
direction of travel for the duration of the flight
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Swenson
4 Differential Equations of Motion
The equations of motion for an object in flight are differential
equations. [1] Solving differential equations can be done analytically or
numerically. In the case of this study, solving these equations analytically
is impossible due to the complexity of the equations. Our equations
include a nonlinear drag term, making that particular equation a
second-order differential equation, which is impossible to solve
analytically. We used Microsoft Excel to solve these equations numerically
using Eulers Method.
4.1 Eulers Method and Excel
Eulers method is a first-order Runge-Kutta method for solving
differential equations numerically. This is done by approximating the
future value of a function by adding the previous value of the function to
the derivative of that function multiplied by a time step. Doing this
repeatedly allows Eulers method to approximate values for differential
equations. In Microsoft Excel, we created columns for each position
function (x, y, and zdirections) as well as for each velocity function
the derivative of each position function as well as a column each for
the drag force on the ball and the time step (See Figure 4). Each row in
the spreadsheet estimates the next value by multiplying the previous
derivative by the time step and adding this to the previous value (for each
direction of motion).
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4.1 Eulers Method and Excel Swenson
Figure 4: Our Excel spreadsheet for Eulers Method showing the columns for each position,
velocity, and acceleration function.
4.1.1 Field Goals
Possibly the most iconic portion of the kicking game is the field goal.
Worth three points, a field goal is often the winning play in many tight
games. Using the differential equations of motion we can now effectively
model a field goal from kick to landing. In order to do this a few values
must be chosen. The geographic location of the kick, the temperature and
humidity of the day, as well as how fast and at which angle the ball is
kicked are all values that must be input into the equations. Equation 12
models the air resistance on a field goal kicked on a summer day in Helena,
Montana with an initial velocity of 126 fts
and a launch angle of 55.
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4.1 Eulers Method and Excel Swenson
Figure 5: A right triangle showing the minimum angle a kicker must kick a field goal to clear
the defensive line.
This launch angle of 24.5 is optimal because it assumes that the
defensive line is held 7yd away by the offensive line. Unfortunately for the
kicker, this is often not the case, the defensive line is many times much
closer than 7 yd, sometimes as close as 5yd from the kicker. If the
defender is 5yd away, and still reaching 10ft high, the kicker must now
kick with an initial = 33.7. This does not seem like much of an angle,
but with the pressure of the defense looming at him, the kicker will try to
launch the ball as high as possible, especially on shorter field goals. For a
PAT the kicker is only 20yd from the goal post, which in terms of field
goals is not far at all. We will start by graphing the trajectory of a PAT
kicked at 50, with initial velocity of 90 fts. Figure 6 shows this trajectory.
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4.1 Eulers Method and Excel Swenson
Figure 6: A PAT kicked at 90 fts
at an angle of 50.
As is often the case on longer field goals, the kicker will feel the
pressure to kick the ball farther, and will consequently lower the launch
angle of the ball. This feels to a kicker like it should increase the distance
the ball travels, however this is not true. As can be seen by a plot of the
trajectory of a ball launched at 90 fts
at an angle of 35 in Figure 7, the ball
covers no more distance than does the ball in Figure 6.
Figure 7: A field goal kicked at 90 fts
at an angle of 35.
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4.1 Eulers Method and Excel Swenson
There is actually a major drawback to kicking the ball at a lower
angle; the ball kicked at 50 reaches a max height of almost 60ft while the
ball kicked at 35 only reaches a maximum height of 30ft. This is a
problem because the goal post crossbar is 10ft high, not to mention the
defensive linemen trying to block the kick jumping at least 10ft into the
air. Because of the challenges, height is very much as important as
distance when kicking field goals.
4.1.2 Kickoffs
The very first play of any football game, and the only guaranteed play
for a kicker is the kickoff. One kicker and ten other players line up at the
35 yard line and race down the field to crush whoever has the football.
Before any racing or crushing may occur however, the kickoff itself must
take place. The kickoff is different from the field goal in many respects.
The time pressure on a kickoffis substantially less, as there is no other
team rushing for the ball. The kicker has 25 seconds to place the ball, take
his steps, and kick the ball. Most kickers take anywhere from 7 to 10 steps
before kicking the ball, many more than the 3 that are taken before a field
goal. As a result, the kicker is at a run when he kicks the ball, adding to
the balls initial velocity. Another difference between the kickoffand the
field goal is that the kickoffis done offof a 2in kicking tee, rather than off
of the ground. This allows for more power to be put into the kick as well,
because there is a decreased risk of kicking the ground instead of the ball.
The main similarities between the kickoffand the field goal are the
mechanics of the kick itself (i.e. the part of the foot contacting the ball,
the type of leg swing, and the way the ball rotates in the air.) A kickoffis
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4.1 Eulers Method and Excel Swenson
Figure 9: A punt kicked at 80 fts
at an angle of 60.
One other, more strategic difference between a punt and the other
types of kicking is the goal of the kick. The main goal of a punt is the
hang time, the total amount of time before the ball hits the ground. A
longer hang time allows the punters teammates to race down the field and
stop the recipient of the punt before he is able to advance the ball. As a
result, punts are often kicked at much greater angles than kickoffs for
example, the main goal of which is to kick the ball as far as possible. The
punter is often worried about kicking the ball too far actually, because
many times the punter could kick the ball farther than the end of the field,
resulting in a touchback (good field position) for the other team.
Therefore, punts usually fly much higher and sometimes shorter than
kickoffs and field goals.
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Swenson
5 Wind
A major factor affecting the trajectory of any object in flight is the air
resistance. Often negated, this factor has a substantial impact on the
trajectory of the object. The next major factor that influences the flight of
a projectile is wind. In this study, we have looked at the force due to air
resistance and the effects that drag forces have on a football. As any
kicker knows, wind makes a huge difference in the trajectory of a ball. It is
such a big factor that teams often decide which half of the field to defend
based on the direction of the wind. In this study we will examine the
eff
ects of crosswinds, tailwinds, and headwinds on a kicked football.
5.1 Crosswinds
Possibly the most critical moment in the kicking game is the field goal.
This is also when the wind often plays the most damaging role. A
crosswind during a last-second field goal can be the difference between
winning a championship or leaving the field in defeat. In our Excel
spreadsheet, columns were added for velocity, position, and the force due
to wind in the xdirection. The force due to the crosswind was
calculated using Equation 14.
Fcrosswind=1
2CADv2 (14)
=1
2(0.6)(0.41)(0.0645)v2
= 0.0079v2
In this equation it is important to notice that the values chosen for the
parametersCand Aare those for a football traveling broadside-first. This
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5.1 Crosswinds Swenson
is because from the perspective of a crosswind, the football is always
showing its broadside during a field goal or kickoff. Whether traveling in a
spiral or tumbling end-over-end, the footballs long axis is constantly
exposed to the crosswind. The value chosen for D is D= 0.0645, the air
density in Helena, MT on an average summer day. The v2 in the function
represents the velocity of the wind with respect to the ball. To calculate
this, our Excel spreadsheet finds the difference between the velocity of the
wind and the velocity of the ball in the direction of the crosswind.
Squaring this difference gives us v2, or the square of the winds velocity
with respect to the ball. Our spreadsheet calculates the position of the
football in the xdirection by using Eulers method, multiplying the
previous derivative (velocity in the xdirection) by the time step (.2
seconds). The spreadsheet calculates the velocity in the xdirection in
much the same way. Multiplying the previous velocity by the quotient of
the force due to the crosswind over the the velocity of the wind, dividing
the product by
1
2CAD, and finally multiplying by the time step gives us
the change in velocity; adding this change in velocity to the previous
velocity gives us the new velocity. This is again Eulers method, and Excel
repeats these calculations for every time step. Figure 10 shows the
position of a football in the xdirection. This figure models a field goal
kicked at 90 fts
at an angle of 50 with a crosswind of 20 fts
=11.63mph.
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5.2 Headwinds Swenson
Figure 11: The movement in the xdirection of a football kicked at 90 fts
at an angle of 50
in a 10mph crosswind in the positive direction.
Our model calculates that a field goal kicked at 90 fts
at an angle of 50
in a 10mph crosswind will be displaced 9.6ft, which is substantially less
than the 16ft displacement caused by a 16mph wind. A wind of 10mph will
push the ball, however a kicker can only change his aim by a few degrees
to keep the ball on target for a valid field goal. Any crosswind with a
velocity greater than 10mph will cause a kicker to need to drastically alter
his aim from dead-center on the goalpost in order to make the field goal.
5.2 Headwinds
Possibly the most frustrating type of wind to a kicker is the headwind.
In any type of kick, a headwind will dramatically alter the trajectory of
the ball. Kickoffs must be kicked at a lower angle because the wind lifts
the ball high into the air and stalls forward movement, causing the ball to
drop nearly straight down onto the field, and often far short of the kickers
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5.2 Headwinds Swenson
From this figure it can be seen that the trajectory is less parabolic and
seems almost squished. This squishing is due to the air resistance
caused by the headwind exerting a force on the ball, effectively working to
push the ball back to the kicker. Figure 13 demonstrates the effects of a
headwind of double the velocity, 13.6mph.
Figure 13: The trajectory of a kickoffaffected by a headwind of 20 fts
= 13.6mph. The ball
was launched at 120 fts.
Figure 13 illustrates an even larger departure from the
characteristically parabolic shape of the trajectory. This stronger
headwind has a more severe effect on the trajectory, causing a more
pronounced squishing. This effect is why kickers often decrease the
launch angle of their kicks in a headwind, to keep the ball from
encountering such a strong effect due to the drag force. A decrease of 5 in
the initial launch angle results in an additional 5.8ft of maximum range
down field, while increasing the launch angle from 35 to 45 decreases the
maximum range of the ball by 24ft, or 8yd. In a headwind of 16.3mph, a
launch angle of 30 provides a much better maximum range than does the
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5.3 Tailwinds Swenson
45 angle that typically provides maximum range to a projectile following
parametric equations.
5.3 Tailwinds
The third and most beloved type of wind for a kicker is the tailwind.
Adding to the velocity in the ydirection, a tailwind increases a kickers
range and aids with kickoffs particularly. For example many college kickers
routinely place the ball within the opposite 5yd line on a kickoff, and a
solid tailwind will add to that range, often boosting the ball into the end
zone for a touchback. It is a goal of this study to discover just how much a
tailwind adds to the maximum range of a kickoff. Tailwinds are also very
beneficial to a punter. With a typically longer hang time, the punt
provides a longer opportunity for the wind to cause an effect on the
trajectory of the ball. Much like the tailwind we adjusted the velocity of
the ball in the ydirection, but instead of subtracting from this
component of the balls velocity, we added to it, increasing the velocity in
the ydirection and consequently extending the maximum range of the
football. Figure 14 illustrates the impact of a 10 fts
tailwind on a kickoff
launched at 120 fts
= 6.8mph at an angle of 35.
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6.3 Future Research Swenson
that this model contains is the fact that the model does not address the
way that wind can scoop a ball upwards on a kickofffor example. We
have assumed that the forces due to air resistance and wind affect the ball
uniformly over the entire leading surface of the ball, causing a deceleration
of the ball in the direction opposite that of motion.
6.3 Future Research
With more time we would like to see this project furthered through
exploration of more precise calculations of air density and the coefficient of
drag for a football. The air densities in this study were calculated using an
online air density calculator, but there are many factors that go into
determining the air density. This is a possible source of error, and more
research into this area would improve this models accuracy. There is no
published drag coefficient for a football, and we would like to explore
calculating a coefficient of drag on a football. In this study the coefficients
of drag for an ellipsoid were used, which are close approximations.
However these coefficients are just that: approximations. More time, a
wind tunnel, and more study of aerodynamics would enable us to derive a
coefficient of drag for a football based on its exact shape and the texture
of the materials used in the football. While our approximations are
effective, finding a more accurate drag coefficient Cwould greatly improve
the accuracy of our model. Another area that we would like to spend time
researching further is the mechanics of the kick itself; the specific physical
events that take place directly before and during the kick. This would help
us to be able to better represent the capabilities of an average kicker,
again improving the accuracy of our model.
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6.4 Summary Swenson
6.4 Summary
Regardless of its weaknesses, this model effectively and appropriately
models the trajectory of a kicked football. While there is room for
improvement and further research, this model takes into account many
factors often ignored and uses them to find an accurate and effective
method to model the trajectory of a ball kicked or punted through the air
in any geographic location. Complimenting this models weaknesses are its
many strengths which make this model effective.
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REFERENCES Swenson
References
[1] Brancazio, Peter J. (1987). Rigid-Body dynamics of a football
American Journal of Physics, 55. 415-420.
[2] Brancazio, Peter J. (1985). The Physics of Kicking a Football. The
Physics Teacher, 53. 403-407.
[3] Denysschen, (n.d.). Air Density Calculator, Web. Accessed 2 Feb. 2013,
Available at http://www.denysschen.com/catalogue/density.aspx
[4] Gay, Timothy. (2004). The Physics of Football. New York:
HarperCollins.
[5] Rouse, H. (1946). Elementary Mechanics of Fluids. Wisconsin, J.
Wiley.
[6] The Weather Channel. (2012).Monthly Averages for Denver, CO.
Web. Accessed 11, Nov. 2012, Available at
http://www.weather.com/weather/wxclimatology/monthly/graph/USCO0105
[7] Weather Spark Beta. (2012).Humidity Averages. Web. Accessed 11,
Nov. 2012, Available at
https://weatherspark.com/averages/30486/Helena-Montana-United-States