4/2/2013
Football in Flight: A study of the math and physics of the trajectory of a kicked football
Ryan Swenson April 2, 2013
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 team’s 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.
2
CONTENTS
Swenson
Contents 1 Introduction
6
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Models
7
2.1
Vertical Motion . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Horizontal Motion . . . . . . . . . . . . . . . . . . . . . . . .
8
3 Drag Forces
9
3.1
Coefficient of Drag . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3
Air Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4
Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4.1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Differential Equations of Motion 4.1
9
15
Euler’s 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
5.3
Tailwinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Conclusion
30
3
CONTENTS
Swenson
6.1
Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2
Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3
Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4
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 Trajectory . . . . . . . . . . . . . . . . . . . . . . 19
7
Field Goal Low Trajectory . . . . . . . . . . . . . . . . . . . . 19
8
Kickoff Trajectory . . . . . . . . . . . . . . . . . . . . . . . . 21
9
Punt Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . 22
10
Crosswind of 20 fts Affecting a Field Goal . . . . . . . . . . . . 25
11
10mph Crosswind on a Field Goal . . . . . . . . . . . . . . . 26
12
Effects of a 6.8mph Headwind on a Kickoff
13
Effects of a 13.6mph Headwind on a Kickoff . . . . . . . . . . 28
14
Effects of a 6.8mph Tailwind on a Kickoff . . . . . . . . . . . 30
5
. . . . . . . . . . 27
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
6
Swenson
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
g Z = − t2 + h 0 2
(1)
where Z 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.
g Z = − t 2 + v 0 + h0 2
7
(2)
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 θ = 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 θ = 90◦ . Motion towards the goal posts Vy = v0 cos(θ)
(3)
Vz = v0 sin(θ) − gt
(4)
And motion upwards
These equations use the sine and cosine of the launch angle θ to determine how much of the initial velocity is in the y− and z− directions, and therefore the y− and z− componenets of the velocity. [1] For example, an object launched at a 60◦ angle at a velocity of 75 fts will have horizontal and vertical velocity components as follows:
vx = 75cos(60) = 37.5
ft s
(5)
and vy = 75sin(60) − 32t = 64.95
8
ft s
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 object’s 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
1 Fdrag = CADv 2 2
(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.
9
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 kickoff or field goal, the average of these two coefficients will be used. Ctumbling =
3.2
0.1 + 0.6 = 0.35 2
(7)
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 = πr2 = 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 = 0.33ft2 2
(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.
10
3.3 Air Density
3.3
Swenson
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 slugs . [7] In Denver, CO, at the ft3 “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 . [6] Although ft3 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 1 Fdrag = CADv 2 2 1 Fdrag = (0.1)(0.25)(0.064)v 2 2 = 0.0008v 2
11
(9)
3.4 Force Equations
Swenson
And for a football traveling end-over-end 1 Fdrag = CADv 2 2 1 Fdrag = (0.35)(0.33)(0.064)v 2 2
(10)
= 0.0037v 2 It is important to note that in Equation 10 we have taken the average of the drag coefficients for a ball flying broadside-first and a ball flying nose-first to find a coefficient C for a ball rotating end over end, as we did with the cross-sectional areas. This coefficient 12 CAD ranges from 1 2 CAD
= 0.0008 for a ball flying in a spiral to
1 2
∗ C ∗ A = .0037 for a ball
tumbling end-over-end. We can then measure the ratio of the drag force for a ball tumbling end-over-end (i.e. a field goal or kickoff) to a ball flying in a spiral.
Fdrag (tumbling) .0008 = = 4.6 Fdrag (spiraling) .0037
(11)
From this we determine that the force due to the drag on the football is 4.6 times greater for a football tumbling end-over-end than for a ball traveling in a spiral fashion. This shows how advantageous it is for a punter to punt the ball in as tight a spiral as possible, rather than an end-over-end “knuckleball” kick. This reduces the drag on the ball caused by the air around it, and will allow the punter to punt the ball over a greater distance.
3.4.1
Results
The differences in air resistance between footballs kicked in a spiral, end-over-end, or long axis perpendicular to the flight path are enormous.
12
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◦ .
13
3.4 Force Equations
Swenson
Figure 2: The trajectory of a football kicked end-over-end, as in a kickoff or field goal
Finally, a football kicked in such a way that the ball’s 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
14
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 Euler’s Method.
4.1
Euler’s Method and Excel
Euler’s 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 Euler’s method to approximate values for differential equations. In Microsoft Excel, we created columns for each position function (x−, y−, and z− directions) 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).
15
4.1 Euler’s Method and Excel
Swenson
Figure 4: Our Excel spreadsheet for Euler’s 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◦ .
16
4.1 Euler’s Method and Excel
Swenson
1 Fdrag = CADv 2 2 1 = (0.35)(0.33)(0.0645)v 2 2
(12)
= 0.0008v 2 = 0.0008(1262 ) = 12.7lb A typical field goal is kicked in an end-over-end fashion, and with a steeper angle than a kickoff is typically kicked. This is because when kicking a field goal, the kicker is not only trying to launch the ball up and over the cross bar, he is also attempting to get the ball over the defensive line, who is doing their best to knock the football out of the air. College and NFL kickers kick the ball from 7 to 7.5 yd behind the line of scrimmage, which is typical football strategy at any level. We will assume that the defensive line does not get any closer than the line of scrimmage until after the ball has been kicked. Assuming that a typical defensive lineman can reach a height of 10ft when jumping, the kicker must kick with an angle of at least 24.5◦ just to ensure that the ball clears the defensive lineman. This estimate that a lineman can reach a height of 10ft is probably a conservative one since many linemen are more than 6ft 4in tall and can reach higher than 10ft, meaning that the minimum launch angle required may be even greater.
tan(θ) =
10ft = .47 21ft
θ = arctan(.47) θ = 24.5◦
17
(13)
4.1 Euler’s 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 θ = 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.
18
4.1 Euler’s Method and Excel
Swenson
Figure 6: A PAT kicked at 90
ft s
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◦ .
19
4.1 Euler’s 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 kickoff is 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 ball’s initial velocity. Another difference between the kickoff and the field goal is that the kickoff is done off of 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 kickoff and 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 kickoff is
20
4.1 Euler’s Method and Excel
Swenson
an end-over-end kick just as a field goal is. Figure 8 demonstrates a kickoff that was kicked at 120
ft s
at an angle of 35◦ .
Figure 8: A kickoff kicked at 120 fts at an angle of 35◦ .
4.1.3
Punts
The final type of kicking we will explore is the punt. The punt is a defensive kick used on fourth down to deliver the ball to the other team, as deep on their side of the field as possible. Substantially different than the kickoff or the field goal, the punt is not kicked from the ground as are the other types of kicks we have explored. The punter holds the ball in his hand – which he caught from the snapper – and drops it, swinging his leg up to kick the ball as it falls. The way the ball behaves in the air is also much different, rather than tumbling end-over-end like a field goal, the optimal punt spirals through the air like a thrown pass. This dramatically reduces the drag forces on the football and adds significant distance to the trajectory. Figure 9 illustrates a punt kicked at 80 spiraling rather than tumbling.
21
ft s
at an angle of 60◦ ,
4.1 Euler’s 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 punter’s 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.
22
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 effects 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 x− direction. The force due to the crosswind was calculated using Equation 14.
1 Fcrosswind = CADv 2 2 1 = (0.6)(0.41)(0.0645)v 2 2
(14)
= 0.0079v 2 In this equation it is important to notice that the values chosen for the parameters C and A are those for a football traveling broadside-first. This
23
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 football’s 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 v 2 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 v 2 , or the square of the wind’s velocity with respect to the ball. Our spreadsheet calculates the position of the football in the x− direction by using Euler’s method, multiplying the previous derivative (velocity in the x− direction) by the time step (.2 seconds). The spreadsheet calculates the velocity in the x− direction 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 12 CAD, 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 Euler’s method, and Excel repeats these calculations for every time step. Figure 10 shows the position of a football in the x− direction. This figure models a field goal kicked at 90 fts at an angle of 50◦ with a crosswind of 20 fts =11.63mph.
24
5.1 Crosswinds
Swenson
Figure 10: The movement in the x− direction of a football kicked at 90 fts at an angle of 50◦ in a 20 fts = 13.63mph crosswind in the positive direction.
As can be seen by Figure 10 the velocity of the ball in the x− direction increases with time. For this particular kick, the displacement of the ball due to the crosswind is over 16ft as the ball lands, after 3.9seconds of flight. This may seem like too large a change in position in the x− direction, however college and NFL goalposts are 18ft and 6in wide. This means that the displacement of the ball due to the crosswind is not enough to move the ball from one side of the goalposts to the other and therefore miss the field goal. However, this crosswind is severe enough for the kicker to need to aim at (or near) the far left post in order to make the field goal in spite of the crosswind’s displacement of the ball to the right. Figure 11 demonstrates the effects of a 10mph crosswind in the positive direction on a field goal kicked at 90 fts at an angle of 50◦ .
25
5.2 Headwinds
Swenson
Figure 11: The movement in the x− direction 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 kicker’s
26
5.2 Headwinds
Swenson
ideal target. Unlike a crosswind, the effects of a headwind depend upon the type of kick. A spiraling punt for example, has less surface area for the wind to exert force upon whereas a tumbling kick from a field goal or a kickoff has a greater (average) area for the wind to exert force upon. In Excel, we used a very similar computation to calculated the force due to the headwind as we did the force due to the crosswind. One difference was that instead of using the values for C and A for a ball flying broadside-first, we used the values of C and A for a tumbling ball. Another difference is that instead of adding to the velocity of the ball in the x− direction, we subtracted from the velocity of the ball in the y− direction. This has the effect of retarding the velocity of the ball in the direction of the kick, and decreases the maximum range of the football. Figure 12 illustrates the effects of a 6.8mph headwind on a tumbling kickoff kicked at 120 fts and at an angle of 35◦ .
Figure 12: The trajectory of a kickoff affected by a headwind of 10 fts = 6.8mph. The ball was launched at 120 fts .
27
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 kickoff affected 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
28
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 y− direction, a tailwind increases a kicker’s 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 y− direction, but instead of subtracting from this component of the ball’s velocity, we added to it, increasing the velocity in the y− direction 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◦ .
29
Swenson
Figure 14: The trajectory of a kickoff affected by a tailwind of 10 fts = 6.8mph. The ball was launched at 120 fts .
A tailwind of 6.8mph gives a kickoff a significant boost in maximum range, in this case that extension of maximum range is over 51.1ft. This is a very good kickoff – even for an NFL kicker – to begin with, and the addition of the small tailwind adds 17yd to the kick, guaranteeing a touchback. This demonstrates that tailwinds do not have to be drastic to have drastic effects on a football. Compared to headwinds, tailwinds do not “squish” the latter half of the trajectory of the football; rather tailwinds “stretch” the trajectory, elongating the ball’s descent to the ground.
6
Conclusion Many factors are involved in the trajectory of a projectile. Launch
angle and initial velocity are just the beginning and are complicated with the inclusion of the football’s odd shape, force due to air resistance, and
30
6.1 Strengths
Swenson
any potential forces due to wind. This study included these other factors often ignored by math and physics classes. Introducing a second-order differential equation made the system impossible to solve analytically and required the use of numerical analysis, namely Euler’s Method, which is a second-order Runge Kutta method for numeric integration. The use of Microsoft Excel and Euler’s method allowed us to solve the differential equations of motion regardless of the squared drag coefficient.
6.1
Strengths
As with any mathematical model, ours has strengths and weaknesses. Some strengths are that this model includes air resistance and wind, that it includes explorations of kickoffs, field goals and punts, and that it uses a first-order Runge Kutta method (Euler’s method) for solving differential equations numerically. In this study we used a coefficient for drag for an ellipsoid and were able to use air densities realistic for different geographic areas. The resulting trajectories are effective approximations of the flight path of a football kicked in a spiral or tumbling fashion.
6.2
Weaknesses
Perhaps more interesting are this model’s weaknesses and sources for improvement through further work and research. For example, in this model we assume that a football rotating end-over-end spends equal amounts of time traveling nose-first and broadside-first. This may be true for some speeds of rotation, but for a ball rotating more slowly the area exposed to the force due to air resistance may in fact not be the average of the broadside and cross-sectional areas of a football. Another weakness
31
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 kickoff for 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 model’s 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 C would 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.
32
6.4 Summary
6.4
Swenson
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 model’s weaknesses are its many strengths which make this model effective.
33
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
34
