Parametric Equations: A Hole in One!

Author:
Robert Reed
Topic:
Vectors

Introduction

A golfer is practicing a trick shot that involves hitting a golf ball off of a cliff and landing it directly into a hole. We are going to investigate how the angle and strength of the hit affect how far the ball goes. Our goal will be to precisely describe the golf ball's motion using parametric equtions so that we can help our golfer get a hole-in-one every time. Begin by watching the brief introduction below, and then start working your way through the five tasks. For each task, record your results in the input box provided. Tasks 1-3 require no prior knowledge, but Tasks 4 and 5 do. Tasks 4 and 5 rely on previous experience with the following ideas: - Distance as a function of rate and time - The quadratic model for vertical motion - Decomposition of a vector into its vertical and horizontal components - Modeling the path of a projectile using parametric equations - Eliminating the parameter in a set of parametric equations - Using a graphing calculator to solve for unknown values If you are feeling lost, try to use the hints and videos at the bottom of this page as a resource to get you moving in the right direction again. Have fun!

An Introduction to the Activity

Common Core State Standards addressed in this activity

HS.MP.1 - Make sense of problems and persevere in solving them. HS.MP.2 - Reason abstractly and quantitatively. HS.MP.3 - Construct viable arguments and critique the reasoning of others. HS.MP.4 - Model with mathematics. HS.MP.5 - Use appropriate tools strategically. HS.MP.6 - Attend to precision. HS.MP.7 - Look for and make use of structure. HS.N.VM.3 - Solve problems involving velocity and other quantities that can be represented by vectors. HS.A.CED.2 - Create equations in two or more variables to represent relationships between quantities. HS.A.CED.4 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. HS.A.REI.1 - Construct a viable argument to justify a solution method.

Task 1: Exploring

Use the sliders to adjust the angle and velocity of the hit. Pay attention to how the trajectory of the ball changes as you manipulate these values. Record your observations in the space below the activity.

Describe how the trajectory of the ball changed as you manipulated the angle and velocity of the hit.

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Task 2: Experimenting

Use the sliders to change the angle and velocity of the hit. Find three different combinations that appear to give you a direct shot into the hole. As an added challenge, try to find a velocity that will allow you to make a direct shot into the hole with two different hit angles. Record your answers in the space below the activity.

Record your three answers below in the format [Angle, Velocity] and separate each answer with a line break.

Task 3: Approximating a Solution

Given a hit angle of 30, approximate the velocity that will result in a direct shot into the hole. You should see the red ball disappear entirely if you find the correct value. Record your results below.

What velocity resulted in a direct shot into the hole when the hit angle was 30?

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Task 4: Finding and Using a Model

It would be nice to know if a specific hit angle and velocity results in a direct shot into the hole without having to guess. In this scenario the golf ball is resting on a tee at a height of 12 feet above the hole, which is located 33.435 feet away horizontally. The golfer swung and launched the ball with an initial velocity of 55 f/s at an angle of 75. We would like to determine if this results in a direct shot into the hole, but in order to do so we'll need to write a set of parametric equations that models the location of the ball at time t seconds.

Organizing Information

In order to write our model we'll need to identify the important information from the passage above. In the space below, record any important information that you think we may need in solving this problem.

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Writing a Model

Use the information above to come up with a set of parametric equations that will model the path of the ball. Feeling stuck? Take a look at the video below for a quick refresher on how to do this. Note that the video provides a generalized solution to this problem, but it will still be up to you to apply it to this particular scenario.

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Modeling Projectile Motion: Review

The exact location of the hole is (33.435, 0). Use your model from the previous question to determine if the shot was in fact a direct shot into the hole. Explain how you determined your answer and provide any supporting evidence used in your explanation.

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Test your results!

Task 5: Review and Extensions

The following is a mixture of review and extension problems that relate to the material investigated in this problem set. You may use any resources at your disposal to help you with the problems. All answers should be rounded to the thousandth place. If you are stuck on a problem, scroll down to access hints and video walkthroughs for each problem.

Question 1

The golfer hits the ball at an angle of 71.652 with an initial velocity of 40 f/s and sinks a direct shot into the hole. Given that the ball starts at a height of 12 feet and the hole is located at (33.435, 0), how long does it take for the ball to reach the hole? Feeling stuck? Scroll down for hints and help with this problem.

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Question 2

This time the golfer hits the ball with an initial velocity of 72 f/s at an angle of 84. What is the maximum height above the hole that the ball reaches, and how long does it take to get to that point? Feeling stuck? Scroll down for hints and help with this problem.

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Question 3

In this round the golfer hits the ball at an angle of 87. Determine the velocity of the hit that would result in a direct shot into the hole. Feeling stuck? Scroll down for hints and help with this problem.

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Question 4

A more challenging hole is located at the coordinates (100, 0). If the ball is hit with an initial velocity of 95 f/s, determine the two angles that would result in a direct shot into the hole. Feeling stuck? Scroll down for hints and help with this problem.

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Hints and Videos for Task 5

The following hints should be used as needed. Read only one line of the hint at a time. Once your memory is sparked, try to use your materials from previous units to solve the problem. If you find yourself totally stuck, watch the video walkthrough found in the next section as it will explain one possible approach to solving the problem.

Hints for Question 1

1) We should probably be using the model , . 2) We know that the hole is at a height of . Try to think of a way to use that information combined with one of the equations from the previous hint to find the solution.

Solution for Question 1

Hints for Question 2

1) Try only using the model for the vertical motion of the ball. 2) Try graphing the model for the vertical motion of the ball and identifying the vertex of the parabola. Make sure to interpret your results correctly! 3) Feeling fancy? Remember that the instantaneous vertical velocity of the object is zero when at its maximum height. Hmmmm, smells like a derivative to me!

Solution for Question 2

Hints for Question 3

1) Write the parametric equations that model this situation generically in terms of the unknown initial velocity. 2) You should have found the following: , . Now what can you do? 3) Try to eliminate the parameter to get an equation in and . 4) You should end up with something like this: , which still appears to have too many variables for us to use. What can you do now? What else do you know? 5) Plug the coordinates of the hole into and in the above equation. Now all you need to do is figure out what value of makes that equation true! Hmm, how might you do that? What tools do you have at your disposal? 6) You might want to try graphing the above equation and calculating its x-intercept. Hurrah!

Solution for Question 3

Hints for Question 4

1) This problem is very similar to the previous problem. Write your model generically in terms of and see what you can do from there. 2) Go on, eliminate the parameter already! You know what to do from here! 3) Same approach as before, but this time you have two x-intercepts to calculate. Yee-haw!

Solutuion for Question 4

Closure

Use the space below to summarize your thoughts on the activity. Reflect on the use of mathematical models and parametric equations and their application to solving real-world problems.

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Check Your Answers!

Use the button below to check your answers for the activity. Once the button has been pressed you can scroll through the page and see answers for each question.

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