Analysis of a football punt
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ANALYSIS OF A FOOTBALL PUNT. David Bannard TCM Conference NCSSM 2005. Opening thoughts. Watching St. Louis, Atlanta playoff game, the St. Louis punter punts a ball. At the top of the screen a hang-time of 5.1 sec. is recorded.

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ANALYSIS OF A FOOTBALL PUNT

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Analysis of a football punt

ANALYSIS OF A FOOTBALL PUNT

David Bannard

TCM Conference

NCSSM 2005


Opening thoughts

Opening thoughts

  • Watching St. Louis, Atlanta playoff game, the St. Louis punter punts a ball.

  • At the top of the screen a hang-time of 5.1 sec. is recorded.

  • In addition, I observed that the ball traveled a distance of 62 yds.


What questions might occur to us

What questions might occur to us!

  • How hard did he kick the ball?

  • Asked another way, how fast was the ball traveling when it left his foot?

  • At what angle did he or should he have kicked the ball to achieve maximum distance?

  • How much effect does the angle have on the distance?


More questions

More Questions

  • How much effect does the initial velocity have on the distance?

  • Which has more, the angle or the initial V?

  • What effect does wind have on the punt?


Initial analysis

Initial Analysis

  • Most algebra students have seen the equation

  • Suppose we assume the initial height is 0.

    When the ball lands, h = 0, so we have

  • In other words, a hang-time of 5.0 sec. Would result from an initial velocity of 80 ft/sec


Is this solution correct

Is This Solution Correct?

  • Note that this solution only considers motion in one dimension, up and down.

  • The graph of this equation is often misunderstood, as students often think of the graph as the path of the ball.

  • To see the path the ball travels, the x-axis must represent horizontal distance and the y-axis vertical distance.


Two dimensional analysis

Two dimensional analysis

  • Using vectors and parametric equations, we can analyze the problem differently.

  • We will let X(t) be the horizontal component, I.e. the distance the ball travels down the field, and Y(t) be the vertical component, the height of the ball.

  • Both components depend on the angle at which the ball is kicked and the initial V.


Vector analysis

Vector Analysis

  • The Ball leaves the foot with an initial velocity V0 at an angle q with the ground.


Vector analysis1

Vector Analysis

  • The Ball leaves the foot with an initial velocity V0 at an angle q with the ground.

Initial Velocity V0

q


Vector analysis2

Vector Analysis

  • The horizontal component depends only on V0t and the cosine of the angle.

Initial Velocity V0

q


Vector analysis3

Vector Analysis

  • The horizontal component depends only on V0t and the cosine of the angle.

Initial Velocity V0

q

X(t)=V0t cos q


Vector analysis4

Vector Analysis

  • The horizontal component depends only on V0t and the cosine of the angle.

  • The vertical component combines v0t sinq and the effects of gravity, –16t2.

Initial Velocity V0

q

X(t) = V0t cos q


Vector analysis5

Vector Analysis

  • The horizontal component depends only on V0t and the cosine of the angle.

  • The vertical component combines v0t sinq and the effects of gravity, –16t2.

Initial Velocity V0

Y(t) = –16t2 + V0t sinq

q

X(t) = V0t cos q


Calculator analysis

Calculator analysis

  • In parametric mode, enter the two equations.

  • X(t)=V0t cos q + Wt where W is Wind

  • Y(t)=–16t2+V0t sin q + H0 where H0 is the initial height.

  • However we will assume W and H0 are 0


Initial parametric analysis

Initial Parametric Analysis

  • Suppose that we start with t = 5 sec. and V0=80 ft./sec.

  • We need an angle, and most students suggest 45° as a starting point.

  • These values did not give the results that were predicted by the original h equation.

  • Try using a value of q=90°.


Trial and error

Trial and Error

  • Assume that the kicking angle is 45°. Use trial and error to determine the initial velocity needed to kick a ball about 62 yards, or 186 feet.

  • What is the hang-time?


New questions

New Questions

  • 1) How is the distance affected by changing the kicking angle?

  • 2) How is the distance affected by changing the initial velocity?

  • 3) Which has more effect on distance?


Data collection

Data Collection

  • Collect two sets of data from the class

  • Set 1: Hold the velocity constant at 80 ft/sec. And vary the angle from 30° to 60°.

  • Set 2: Hold the angle constant at 45° and vary the velocity from 60 ft/sec to 90 ft/sec.


Accuracy

Accuracy

  • Accuracy will improve by making delta t smaller. Dt = 0.05 is fast. Dt = 0.01 is more accurate.

  • Do we wish to interpolate?

  • First estimate the hang-time with Dt = 0.1

  • Use Calc Value to get close to the landing place.

  • Choose t and X at the last positive Y.


Use a spreadsheet and or calculator to collect data

Use a Spreadsheet and/or calculator to collect data.

Then analyze the data using data analysis techniques on a calculator


Algebraic analysis

Algebraic Analysis

  • Can we determine how the distance the ball will travel relates to the initial velocity and the angle. In particular, why is 45° best?


Analysis of a football punt

  • X(t) = V0t cos q and Y(t) = –16t2 + V0t sin q

  • When the ball lands, Y = 0, so

  • –16t2 + V0t sin q = 0 or t (–16t + V0 sinq) = 0

  • So t = 0 or V0 sinq/16.

  • But X(t) = V0t cos q


Analysis of a football punt

  • X(t) = V0t cos q and Y(t) = –16t2 + V0t sin q

  • When the ball lands, Y = 0, so

  • –16t2 + V0t sin q = 0 or t (–16t + V0 sinq) = 0

  • So t = 0 or V0 sinq/16.

  • But X(t) = V0t cos q

  • Substituting gives


Analysis of a football punt

  • X(t) = V0t cos q and Y(t) = –16t2 + V0t sin q

  • When the ball lands, Y = 0, so

  • –16t2 + V0t sin q = 0 or t (–16t + V0 sinq) = 0

  • So t = 0 or V0 sinq/16.

  • But X(t) = V0t cos q

  • Substituting gives

  • Using the double angle identity gives


Analysis of a football punt

  • Finally, we have something that makes sense.

  • If V0 is constant, X varies as the sin of 2q, which has a maximum at q = 45°.

  • If q is constant, X varies as the square of V0.


Additional results

Additional results

  • How do hang-time and height vary with q and V0?

  • We already know the t = V0 sinq/16

  • The maximum height occurs at t/2, so


Additional results1

Additional results

  • How do hang-time and height vary with q and V0?

  • We already know the t = V0 sinq/16

  • The maximum height occurs at t/2, so


Final question if we know the hang time and distance can we determine v 0 and q

Final QuestionIf we know the hang-time, and distance, can we determine V0 and q?

  • Given that when Y(t)=0, we know X(t) and t.

  • Therefore we have two equations in V0 and q, namely

  • X = V0t cos qand 0 = –16t2 + V0t sinq.

  • Solve both equations for V0 and set them equal.


If we know the hang time and distance can we determine v 0 and q

If we know the hang-time, and distance, can we determine V0 and q?

  • Given that when Y(t)=0, we know X(t) the distance and t, the hang-time.

  • Therefore we have two equations in V0 and q, namely

  • X = V0t cos qand 0 = –16t2 + V0t sinq.

  • Solve both equations for V0 and set them equal.


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