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

                                   32

         

          (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

                      32

             

          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°?