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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

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?
slide22
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
slide23
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
slide24
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
slide25
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 q and 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 q and 0 = –16t2 + V0t sinq.
  • Solve both equations for V0 and set them equal.
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