In this module, you will model one drop of a coaster by marking the peak and valley of the drop and then fitting (in height and slope) a cubic polynomial to the marked points. Once you have determined the function, you will then calculate the thrill of the single drop. We provide a downloadable Maple worksheet with commands and explanations.
I. Getting Started
Click the button at the right to open the MAPLE worksheet cubiccoaster1hill.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 Colossus (Module A, page 2).
Enter the x coordinates of your peak point and valley point using the list syntax ( [x1,x2] ) for the xdata variable.
Enter the y coordinates of your peak point and valley point using the list syntax ( [y1,y2] )for the ydata variable.
Enter the slope conditions for your peak point and for your valley point using the list syntax ( [s1,s2] ) for the slopes variable.
III. Connecting Cubic Polynomial
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 that fits the given peak and valley points. A close examination of the commands shows that Maple determines the unknown coefficients 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 cubic polynomial function. Does this match your coaster hill?
IV. Calculation of the Angle of Steepest Descent
Now we must determine the steepest point on the curve (i.e., the coaster drop). In other words, we must determine the minimum value of the derivative on the x interval (determined by the peak and valley points). In order to work with a positive-valued function, we rephrase this as determining the maximum value of the absolute value of the derivative on the x interval.
How do we maximize |f'| on a closed interval? We determine critical points of f' and then compare function values of f' at critical points and endpoints. The Maple commands calculate and then graph f'(x). Then, the critical points of f' are found by solving f"(x) = 0 on the restricted x interval. Finally, we evaluate f'(x) at all critical points and endpoints and choose the maximum absolute value.
What is the x coordinate of the point of steepest descent?
What is the relation between the point of steepest descent and the peak and valley points?
Is this relation true for all functions?
How does the slope at the steepest point compare to your previous work from module A?
To determine the angle of steepest descent, we must convert slope measurement into angle measurement. Using a right triangle, we see that the radian measure of the angle of steepest descent is given by the arctangent of the slope.
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 the drop based on the definition (see page 1 or page 2).
VI. Observation and Generalization
Repeat using collected data points (from Module A, page 2) for the single drop of the Steel Dragon.
Keeping in mind the coaster restrictions, experiment with several different peak and valley combinations. Keep a record of your results.