Football In Flight.pdf

7/25/2019 Football In Flight.pdf

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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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Swenson

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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Swenson

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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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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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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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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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.

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[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

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