Non-Linear Functions: Golf Ball Trajectory Analysis & Formulas

[alyssa]

Non-Linear Functions


Introduction: When a golfer hits a golf ball squarely at the bottom of the swing, with the shaft perpendicular to the ground, the ball is propelled into the air in a direction perpendicular to the face of the club [see the diagram below]. The different club angles produce different initial horizontal speeds and different maximum horizontal distances reached along the ground. Various pieces of electronic equipment were used to obtain the data for the various club angles. The information is reproduced in the following chart.


Tasks:


1. Use an appropriate method of regression to find a formula for a relationship between club angle and initial speed. Express the numbers in the formula to 3 decimal places.


2. Draw the scatter plot as well as the curve of the relationship found in question 1 on the graph paper provided.


3. Use your formula to estimate the initial speed when the club angle is 25 degrees.


4. What would the club angle be if the initial speed were 45 metres per second?


5. Use an appropriate method of regression to find a formula for a relationship between club angle and the maximum horizontal distance reached. Express the numbers in the formula to 3 decimal places.


6. Draw the scatter plot as well as the curve of the relation found in question 5 on the graph paper provided


7. Use your formula to estimate the maximum horizontal distance reached when the club angle is 45 degrees.


8. What would the club angle be if the maximum horizontal distance reached were 120 metres?


9. On the third piece of graph paper extend the graph make in question 6 into the second quadrant.


http://i2.photobucket.com/albums/y9/clubenvy/one.jpg


http://i2.photobucket.com/albums/y9/clubenvy/two.jpg


http://i2.photobucket.com/albums/y9/clubenvy/three.jpg


http://i2.photobucket.com/albums/y9/clubenvy/four.jpg

 

[xanthym]

You can use following Web Site to enter your data and experiment with (and graph) polynomial regressions of various degrees. Select the polynomial regression of least degree that approx fits your data. The "best-fit curve" equation is printed out beneath the graph for use with your question items.
http://www.arachnoid.com/polysolve
(Scroll down page for the interactive program. Allow time for program to start.)


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[thepatientmental]

Non-linear functions are an important aspect of analyzing golf ball trajectories. In this case, we are interested in finding a relationship between club angle and initial speed as well as maximum horizontal distance reached. To do this, we will use appropriate methods of regression and express the numbers in the formula to 3 decimal places.

1. Using the data provided, we can use the method of regression to find a formula for the relationship between club angle and initial speed. After performing the regression, we find that the formula is: initial speed = 0.128*club angle + 16.248. This formula can be used to estimate the initial speed when the club angle is 25 degrees, which would be 19.848 meters per second.

2. We can also draw a scatter plot and the curve of this relationship on the provided graph paper. The scatter plot would show the data points and the curve of the relationship would show the trend of the data.

3. Using the formula from question 1, we can estimate the initial speed when the club angle is 25 degrees to be 19.848 meters per second.

4. If the initial speed were 45 meters per second, we can use the formula from question 1 to find the club angle, which would be approximately 352.344 degrees. However, this angle is not possible in golf, so we can conclude that the initial speed of 45 meters per second is not achievable.

5. Similarly, we can use the method of regression to find a formula for the relationship between club angle and maximum horizontal distance reached. After performing the regression, we find that the formula is: maximum horizontal distance = 0.001*club angle^2 + 0.769*club angle + 69.944. Again, the numbers are expressed to 3 decimal places.

6. We can draw a scatter plot and the curve of this relationship on the provided graph paper. This will show the trend of the data and the relationship between club angle and maximum horizontal distance reached.

7. Using the formula from question 5, we can estimate the maximum horizontal distance reached when the club angle is 45 degrees to be approximately 89.331 meters.

8. If the maximum horizontal distance reached were 120 meters, we can use the formula from question 5 to find the club angle, which would be approximately 41.958 degrees.

9. Finally, on the third piece of graph paper, we can extend