Roller Coaster Track Design Assignment

Assignment Task

Description

For Mathematical Methods, the Mathematical Investigation assessment type consists of one major task. This task will focus predominantly on Differential Calculus. You use your knowledge and understanding of different applications of Differential Calculus to the design of a Rollercoaster.

Concepts and Techniques

CT1 - Knowledge and understanding of concepts and relationships

CT2 - Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts

CT3 - Application of mathematical models

CT4 - Use of electronic technology to find solutions to mathematical problems

Reasoning and Communication

RC1 - Interpretation of mathematical results

RC2 - Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations

RC3 - Use of appropriate mathematical notation, representations, and terminology

RC4 -Communication of mathematical ideas and reasoning to develop logical arguments

Part A

The first task is to design an ascent and drop for a section of roller coaster track. 

You will connect two straight stretches of track, and, with part of a parabola, where x and are measured in metres.

By studying photographs of roller coasters you decide to make the slope of the ascent  _____ and the slope of the drop  ______. 

For the track to be smooth there can’t be abrupt changes in direction. To ensure smooth transitions between the linear segments and the parabola -  and need to be tangents to the parabola f(x) at the transition points P and Q. 

To simplify the equations, you decide to place the origin at P.  Furthermore, the horizontal distance between P and Q is metres.

Find the equations of the three segments of track indicated below for the given domains, showing all appropriate steps of logic:

Part B

After conducting some research on roller coaster specifications, your task is to design a roller coaster and specify the equations for your roller coaster track from the starting point (entrance on the left) to its finish point (exit on the right). The three segments of track from Part A must be included in your track.

You must determine equations (minimum of 4) for the section on the left to join L₁ and equations (minimum of 3) from  L₂ to the exit point, ensuring a smooth transition between points-show detailed working, equations, constraints, reasons for the choices etc.

You must carefully consider:

• length and height constraints
• using a range of functions in your design
• consider different techniques for generating the functions
• Using FX Graph, Geogebra or similar software, draw a detailed, fully labelled, smooth graph showing what each section looks like. You must observe the constraints of each section and need to also produce a final graph showing all sections of the roller coaster track together.
• Include a discussion of any assumptions, limitations and improvements in the design of your roller coaster.

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