ANALYSIS OF A FOOTBALL PUNT

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ANALYSIS OF A FOOTBALL PUNT

• David Bannard
• TCM Conference
• NCSSM 2005

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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.

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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?

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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?

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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

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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.

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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.

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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)16t2V0t sinq
q
X(t)V0t cos q
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Calculator analysis

• In parametric mode, enter the two equations.
• X(t)V0t cos q Wt where W is Wind
• Y(t)16t2V0t sin q H0 where H0 is the initial
  height.
• However we will assume W and H0 are 0

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Initial Parametric Analysis

• Suppose that we start with t 5 sec. and V080
  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 q90.

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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?

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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?

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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.

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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.

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Using a Spreadsheet to collect data.
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Algebraic Analysis

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

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• 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

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• 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.

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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

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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 q and 0 16t2 V0t sinq.
• Solve both equations for V0 and set them equal.