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Assignment Task:

Task:

Purpose
Parametric equations may be used to model the movement of particles or objects in two or three dimensions. Some examples of the types of pathways that can be modelled are listed below:

  • Missiles and interception
  • Golf swing
  • Path of a ball in a ball game
  • Trapeze, gymnasts, trampoline work
  • Paths of planets/moons
  • Stunt motorcycles/bicycles
  • Spiral staircases
  • Assessment Type: Mathematical Investigation Task Weighting 20%
  • Task Summary
  • A set of parametric equations such as st=xt=x0+atyt=y0+bt-12gt2 for t?0, illustrate the position of an object in free flight motion. The value of the gravitational acceleration, g, is given as 9.8ms-2.The velocity and acceleration vector could be determined by using differential calculus, a(t)=v'(t)=s''(t)
  • Part 1: Use of parametric equations to model a serve in tennis
  • Let xt=30tyt=3-3t-4.9t2 be a set of parametric equations. set of parametric equations used to model the serve of a ball in a tennis game, where the distances are measured in metres and the time, t, in seconds. The ball is served from the baseline, through the middle of the court. The path of the ball, Px,y is shown in Figure 1. A tennis court is 23.8 metres long with the net in the middle at a height of 0.9 m. The service line is 6.4 m on either side of the net.
  • Figures 1
  • What to do:
  • Find the initial position of the ball and explain its significance to the practical context of the tennis game.
  • Determine the velocity vector of the ball and hence the speed at which the ball was served.
  • Assuming that the ball clears the net, show that the ball lands inside the service area.
  • Find the time t, that the ball would be passing over the net and hence calculate if the serve will clear the net or not.
  • Discuss any assumptions and limitations of this model in relation to factors that exist in real life.
  • Part 2: Investigating parametric equations in sport
  • Using general equations for the horizontal and vertical position of the ball in flight:
  • Find expressions for the initial position and speed of the ball.
  • Use graphing programs or your graphics calculators to investigate the changes to the curve when parameters changes. You might include in your investigation features of the flight of a ball such as angles of launch, time of flight, maximum height and maximum displacement. Discuss your findings and support your discussions with appropriate graphs.
  • Apply your findings on the set of equations given in Part 1, if the serve that they represent must be improved by increasing the initial speed of the ball by up to 5kmh and the height the ball was served by up to 10cm. Vary the parametric equations to reflect these changes. Determine if the ball would still clear the net and also whether it would land within the service area.
  • Part 3: Modelling pathways
  • Consider what modifications you would make to the parametric equations developed in Part 1 and 2 above if different pathways are to be modeled. You can select one of the examples given in the Purpose section, or you can choose a pathway scenario of your own to investigate, which may include a different ball sport game. Throughout the development of your model you should provide detailed explanations of, and reasons for, the modifications you have made to develop the movement of the particle or object that is representing your pathway scenario.
  • Your model of the pathway scenario should be as realistic as possible and throughout the mathematical investigations you should consider limitations of the models you have developed at all stages of the investigation.
  • The focus of this investigation is to discuss, demonstrate and develop the mathematics behind your model. You will need to consider and demonstrate your process when deciding things such as:
  • the kind of equations to use to model the curve of the chosen pathway. This could include, but not limited to, parametric equations, Bezier curves, and/or trigonometric parameterization
  • the accuracy of your calculations
  • You are expected to demonstrate the use of technology in this investigation.
  • You are strongly advised to read the attached performance standards that this task will be assessed against before you begin this task.
  • Assessment Guide
  • In constructing your investigation, you need to use the following structure:
  • Introduction
  • Define the problem that has been presented to you (the ‘what’)
  • Describe clearly and concisely how this task is applicable to your learning and to the real world (the ‘why’- the significance of the task)
  • List all methods you are going to demonstrate to solve this problem, including any use of technology (the ‘how’)
  • Mathematical Calculations & Analysis
  • Follow all the instructions provided with your problem. Provide answers to all questions. Use headings where appropriate.
  • Provide relevant diagrams or graphs that could help illustrate any concepts you are explaining
  • Clearly explain each step in your mathematical reasoning with notes or short sentences. Justify all choices you make when answering questions
  • Analyse and compare the two methods used to determine the volume of your item. Discuss any differences in the outcomes.
  • Discuss the reasonableness and limitations of the mathematical results in your investigation
  • Conclusion: Summarise your findings by:
  • Outlining any observations or findings you have made
  • Clearly describe and illustrate your final results
  • Suggest any alterations you could make to improve the accuracy of your model.
  • Appendices
  • Includes any additional information you would like to include that strengthens your justification or documents process that is not directly essential in your analysis section.
  • You are expected to provide a bibliography for any resources you have used outside of the Specialist Mathematics course during this task.
  • Submission Requirements Modified
  • Format (including text type) Written report ?Word Count / Length Maximum 15 A4 pages
  • Minimum font size 10 Deadlines Date Submission Requirements
  • Checkpoint Due Assessment Design CriteriaWhat you will be assessed on Task Specific Clarification To meet the level descriptors make sure you:
  • Concepts and Techniques
  • Knowledge and understanding of concepts and relationships.
  • Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts.
  • Application of mathematical models.
  • Use of electronic technology to find solutions to mathematical problems.
  • Reasoning and Communication
  • Interpretation of mathematical results.
  • Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations.
  • Use of appropriate mathematical notation, representations, and terminology.
  • Communications of mathematical ideas and reasoning to develop logical arguments.
  • Complete, concise, and accurate solutions to determine the parameters of a serve in tennis and in the chosen scenario.
  • Appropriate selection of techniques to parameters of a serve in tennis and in the chosen scenario.
  • Development and effective application of parametric equations.
  • Efficient use of technology (EFOFEX or other graphing tools and graphics calculators) to represent and investigate features of parametric equations.
  • Clear, efficient and precise interpretation of the mathematical results. All choices of equations used clearly justified.
  • In-depth discussion of any limitations to the model developed or techniques used to determine the pathways of your chosen curve and the reasonableness of the results.
  • Report constructed in the appropriate format. All graphs clearly labelled. Correct notation proficiently used for all calculations.
  • Mathematical arguments clear and easy to follow.
  • Concepts and TechniquesReasoning and CommunicationAComprehensive knowledge and understanding of concepts and relationships.
  • Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts.
  • Successful development and application of mathematical models to find concise and accurate solutions.
  • Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems. Comprehensive interpretation of mathematical results in the context of the problem.
  • Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations.
  • Proficient and accurate use of appropriate mathematical notation, representations, and terminology.
  • Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments.
  • Effective development and testing of valid conjectures, with proof.
  • BSome depth of knowledge and understanding of concepts and relationships.
  • Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts.
  • Some development and successful application of mathematical models to find mostly accurate solutions.
  • Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems. Mostly appropriate interpretation of mathematical results in the context of the problem.
  • Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations.
  • Mostly accurate use of appropriate mathematical notation, representations, and terminology.
  • Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments.
  • Mostly effective development and testing of valid conjectures, with substantial attempt at proof.
  • CGenerally competent knowledge and understanding of concepts and relationships.
  • Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts.
  • Successful application of mathematical models to find generally accurate solutions.
  • Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems. Generally appropriate interpretation of mathematical results in the context of the problem.
  • Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations.
  • Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy.
  • Generally effective communication of mathematical ideas and reasoning to develop some logical arguments.
  • Development and testing of generally valid conjectures, with some attempt at proof.
  • DBasic knowledge and some understanding of concepts and relationships.
  • Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts.
  • Some application of mathematical models to find some accurate or partially accurate solutions.
  • Some appropriate use of electronic technology to find some accurate solutions to routine problems. Some interpretation of mathematical results.
  • Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations.
  • Some appropriate use of mathematical notation, representations, and terminology, with some accuracy.
  • Some communication of mathematical ideas, with attempted reasoning and/or arguments.
  • Attempted development or testing of a reasonable conjecture.
  • ELimited knowledge or understanding of concepts and relationships.
  • Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems.
  • Attempted application of mathematical models, with limited accuracy.
  • Attempted use of electronic technology, with limited accuracy in solving routine problems. Limited interpretation of mathematical results.
  • Limited understanding of the meaning of mathematical results, and their reasonableness or limitations.
  • Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy.
  • Attempted communication of mathematical ideas, with limited reasoning.
  • Limited attempt to develop or test a conjecture.

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  • Uploaded By : Pearl
  • Posted on : October 12th, 2019
  • Downloads : 231

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