Design of a Thrilling Roller Coaster - Module E. Design of a Coaster: Cubic Functions

Author(s):
Patricia W. Hammer, Jessica A. King, and Steve Hammer

In this module, you will model a straight-stretch coaster (several hills) by marking n peak and valley points and then fitting (in height and slope) n - 1 cubic polynomial functions, one to each consecutive pair of marked points. Once you have determined the functions, you will then calculate the thrill of the coaster. We provide a downloadable Maple worksheet with commands and explanations.

I. Getting Started

Click the button at the right to open the MAPLE worksheet 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 the figure at the right.

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 the Greyhound (Module A, page 2).

Enter the x coordinates of your n peak and valley points using the list syntax ( [x1,x2,...,xn] ) for the xdata variable.

Enter the y coordinates of your n peak and valley points using the list syntax ( [y1,y2,...,yn] )for the ydata variable.

Enter the slope conditions for your n peak and valley points using the list syntax ( [s1,s2,...,sn] ) for the slopes variable.

III. Connecting Cubic Polynomial Functions

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

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

Maple shows a plot of the trig functions. Does this match your coaster path?

IV. Calculation of the Angles 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 (page 4), 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 determine safety of the coaster based on the radian measure of the angle of steepest descent. We also calculate the thrill of each drop and of the coaster, based on the definition (see page 1 or page 2).

VI. Assignment - Coaster Design

Repeat this assignment using your recorded peak and valley data points from several hills of The Devil (from Module A, page 2).

Repeat this assignment using your recorded peak and valley data points from several hills of Shivering Timbers (from Module A, page 2).

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

Patricia W. Hammer, Jessica A. King, and Steve Hammer, "Design of a Thrilling Roller Coaster - Module E. Design of a Coaster: Cubic Functions," Loci (February 2005)

JOMA

Journal of Online Mathematics and its Applications