E. Design and Thrill of a Straight Stretch Coaster using Polynomial Functions

In this section, we build a straight stretch roller coaster using n peak and valley points. Students build n-1 cubic polynomials by choosing two consecutive peak/valley points and then following the approach taken in section D) Building a Single Drop Coaster using Cubic Polynomials. 

I.  Getting Started

Click the button to open a MAPLE worksheet entitled cubiccoasters.mws.  If you are given a choice, you should save the file to your hard drive, then navigate to your hard drive and open the file from there.  In the MAPLE worksheet, position your cursor anywhere in the line  [ > restart ;   and press Enter.   Pressing the Enter key executes the MAPLE code on the current line.  The MAPLE restart command will clear all MAPLE variables.   It is important to do this whenever you start a new MAPLE project.   

Now resize your MAPLE and Internet Explorer windows so that you can see them both, side-by-side. Click in either window to make it the active window.  

Your screen should look something like this:

II.  Data Points

First, carefully work through this module using the sample peak and valley points already entered in the Maple worksheet. Then, use your recorded peak and valley data points collected from several hills of the Greyhound (Module A).

Enter the x coordinates of your n peak point and valley point using the list syntax ( [x1,x2,...,xn] ) for the xdata variable.
Enter the y coordinates of your n peak point and valley point using the list syntax ( [y1,y2,...,yn] )for the ydata variable.
Enter the slope conditions for your peak points and for your valley points using the list syntax ( [s1,s2,...,sn] ) for the slopes variable.

III.  Connecting Cubic Polynomials

Now that you have entered the x coordinates, y coordinates and slope conditions, you can work through the Maple worksheet by simply pressing the Enter key on your computer to execute the Maple commands.

In this section, the Maple commands will determine a cubic polynomial that fits each successive pair of peak/valley points.  A close examination of the commands shows that Maple determines the unknown coefficients of the polynomial by simply solving a system of 4 equations (2 conditions at each of the 2 (peak and valley) points) in 4 unknowns. 

Maple shows a plot of the polynomial functions.  Does this match your coaster hill?

IV.  Calculation of the Angle of Steepest Descent/Ascent

Now we must determine the steepest point on the rise and fall of each coaster hill.  We use what we learned in section VI of Module C. Namely, that for these functions the x-coordinate of the point of steepest descent/ascent is the x-coordinate of the midpoint between peak and valley points. This observation will shorten our mathematical calculations.

V.  Safety Restrictions and Thrill Factor

In this section, we simply determine safety of the coaster based on the radian measure of the angle of steepest descent.  We also calculate the thrill of the drop (not of each hill) based on the definition.

VI.  Assignment - Coaster Design

Repeat this assignment using your recorded peak and valley data points from several hills of The Devil (Module A).

Repeat this assignment using your recorded peak and valley data points from several hills of Shivering Timbers(Module A).

Keeping in mind the coaster restrictions, experiment with several different peak and valley combinations. Keep a record of your results.

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