The Mathematics of a Football Kick

(Parametric Equations)

(Adapted from an article in COMAP)

I Problem

The punter on the 1994 Handley State Championship Football

Team, Michael Partlow, was also a calculus student.

The question I posed was "Is it possible to determine the

velocity of the football as it leaves his foot on his

best kicks?"

II. Background

We have already examined the formula s = v₀t - 16 t²

which gives the height s of an object if it is thrown

vertically up in the air with an initial velocity of v₀.

The units are feet (s) and seconds (t).

But a football is not kicked straight up (hopefully), but

rather at an angle.

So we need to revisit some formulas from Precalculus or

Physics. Looking at the diagram at the right,

fill in the following:

| /|

| r/ |

cos A = ____ | / |y

|/A |

-----+--x-+---

sin A = ____ |

so, x = r cos A and

y = r sin A if there is no gravity.

Recall from algebra that distance = velocity x time

so, r = v t

Substituting for r in the formulas above, we get

x = ____________________________

y = ____________________________ - 16 t²

These formulas neglect wind velocity. Note that only the

vertical motion is affected by gravity - that is why

the -16t² is only subtracted from the y-value.

III. Assumptions

We will assume that there is no wind velocity.

We will make the conjecture that the best angle at which to

kick the football to achieve the maximum distance is a

45° angle (we will prove this later)

IV. Solving the Problem

Michael Partlow's best punt went 56 yards (which is

equivalent to 168 feet).

Substituting the values in the two equations above, we get

168 = v t cos 45°

0 = v t sin 45° - 16 t²

Now solve these equations simultaneously for t and v.

SHOW WORK:

t = _______________ (which represents the "hang time")

v = _______________ (which represents the initial velocity)

V. Doing Parametric Equations on the TI-85 Graphing Calculator

A. You must first change some of the settings:

Press MODE

Select NORMAL

FLOAT

DEGREE

RECTC

PARAM

B. To enter the data for the problem above:

1. Type the following:

45 STO A

73.329 STO V

2. Press GRAPH

Press RANGE

tMin = 0

tMax = 5

tStep = .05

------------

xMin = 0

xMax = 200

xScl = 10

------------

yMin = -20

yMax = 60

yScl = 10

3. Press E(t)=

xt1 = V t cos A

yt1 = V t sin A - 16t^2

Press GRAPH

4. Use the TRACE button to find the following:

a. Find x when y = 0: x = _________

b. Find the values of x, y, and t at its highest

point:

x = __________ y = __________ t = _________

VI. Using calculus to solve for the information above:

Determine the time that it takes for the ball to reach its highest point.

Take the formula y = V t sin A - 16t² and

substitute 73.329 in for V and 45° for A.

Then solve for dy/dt and set it equal to 0 (WHY???)

Solve this equation for t (this will give you half of

the "hang time."

SHOW ALL WORK:

VII. To show that 45° is the optimal angle for kicking a football

the farthest distance:

1. First determine a general equation for x when y = 0.

Begin with the parametric equations

y = V t sin A - 16t² and

x = V t cos A

Set y = 0 to obtain 0 = V t sin A - 16t²

Solve for t (SHOW WORK):

Substitute the nonzero value of t in the parametric

equation x = V t cos A

x = ________________________________

Now use the double angle identity (sin2A = 2sinA cosA)

to express x in terms of sin 2A:

x = _________________________________

This says that if V is constant, the distance (x)

varies sinusoidally with the angle.

2. Use calculus to solve for the angle which gives the best

distance:

V²

Take the equation x = ------ sin 2A and find dx/dA

(Remember V is a constant)

Then set dx/dA = 0 to get the maximum distance (why?)

and solve for A:

SHOW WORK:

3. Use a calculator to solve for the optimal angle:

First, select MODE and change back to FUNC

Press GRAPH and select y(x)=

V²

Graph x = ------ sin 2A by entering

y1 = (73.329)^2 sin (2x) / 32

Select RANGE and enter the following:

xMin= 0

xMax= 90

xScl= 5

yMin= 0

yMax= 200

yScl= 10

Please note that on the calculator, the variable x

represents the angle, and the variable y

represents the distance the ball is kicked (the x

value in the formulas above)

Use the TRACE function to determine the optimal angle

to kick the ball.

Give these values of x and y: x = _____________

y = _____________

Also give the following values:

When x = 50° y = ___________

When x = 30° y = ___________

VIII. Additional Explorations:

A. We can modify our parametric equations to

x = V t cos A + W t and y = V t sin A + H where

W is the velocity of a wind blowing with (+) or

against (-) the kicker, and H is the height of the ball

when it is kicked.

How much is the problem affected if the ball is kicked

from one foot off the ground?

B. How does the wind affect the kick? For example, if there

is a wind of 10 ft/sec with the kicker, at what angle

should you kick the ball to maximize the distance?

The equation is x = (V²sin 2A) / 32 + (10V sinA) / 16

C. Suppose you are kicking a field goal. In this case, the

ball must be more than ten feet above the ground to get

over the goal posts. How much distance is lost to

achieve this condition?

D. Tom Dempsey holds the NFL record for kicking the longest field goal (63 Yards). With what velocity must he have kicked the ball if his angle was optimal?

E. How quickly does the ball go up? Suppose a player with

his arms raised can block a punt if it is under seven

feet high. How close to the kicker must he get? What

if the angle of the kick is 55°?