Design of a Thrilling Roller Coaster

by Patricia W. Hammer, Department of Mathematics and Statistics
and Jessica A. King, Department of Computer Science  
Hollins University
and
 Steve Hammer, Department of Mathematics
Virginia Western Community College

with
roller coaster images:
Colossus from www.ultimaterollercoaster.com
Greyhound and The Devil images from Ken's Classic Coaster Postcards
Steel Dragon and Shivering Timbers images from www.coastergallery.com
Texas Cyclone image from www.coasterphotos.com

 

   

In this project, students will complete a series of modules that require the use of polynomial and trigonometric functions to model the paths of straight stretch roller coasters.   These modules involve the mathematical definition of thrill and calculation of thrill for several real coasters (Module A), design and thrill analysis of single drop coaster hills (Modules B and C) and design and thrill analysis of several drop coasters (Modules D and E).  The ultimate goal of this interactive project is successful completion of an optimization problem (Module F) in which students must design a straight stretch roller coaster that satisfies the following coaster restrictions regarding height, length, slope and differentiability of coaster path and that has the maximum thrill (as defined below.)

Roller Coaster Restrictions
  • The total horizontal length of the straight stretch must be less than 200 feet.
  • The track must start 75 feet above the ground and end at ground level.
  • At no time can the track be more than 75 feet above the ground or go below ground level.
  • No ascent or descent can be steeper than 80 degrees from the horizontal.
  • The roller coaster must start and end with a zero degree incline.
  • The thrill of a drop is defined to be the angle of steepest descent in the drop (in radians) multiplied by the total vertical distance in the drop.  The thrill of the coaster is defined as the sum of the thrills of each drop. 
  • The path of the coaster must be modeled using differentiable functions.

Students must use Maple8 (or any later version) to complete the project.  Students must already be familiar with derivatives and their use in determining maximum and minimum function values.  These ideas play a crucial role in the design and analysis of the coasters. 

For the Instructor

To complete this project, student should work through each of the modules given below.

A.  Introduction to Roller Coaster Design -  In this module, students use an interactive coaster window to mark peaks and valleys of real-life coasters and then calculate the thrill of each drop using the above definition.  Be sure to record the x and y coordinates of the peak and valley points and the slope at the steepest point.  You will need this information to complete parts B - E below.

B.  Design and Thrill of One Coaster Drop Using a Trig Function - In this module, students model one drop of a coaster by marking the peak and valley of the drop and then by fitting (in height and slope) a trig function of the form f(x) = Acos(Bx+C)+D to the marked points.  Once the function has been determined, students then calculate the thrill of the single drop.  A downloadable Maple worksheet with commands and explanation is provided.

C.  Design and Thrill of One Coaster Drop Using a Polynomial Function - In this module, students model one drop of a coaster by marking the peak and valley of the drop and then by fitting (in height and slope) a cubic polynomial to the marked points.  Once the function has been determined, students then calculate the thrill of the single drop.  A downloadable Maple worksheet with commands and explanation is provided.   

D.  Design and Thrill of a Straight Stretch Coaster Using Trig Functions - In this module, students model a straight stretch coaster (several hills) by marking peak and valley points and then by fitting (in height and slope) a trig function to each consecutive pair of marked points.  Once the functions have been determined, students then calculate the thrill of the coaster.  A downloadable Maple worksheet with commands and explanation is provided.

E.  Design and Thrill of a Straight Stretch Coaster Using Polynomial Functions - In this module, students model a straight stretch coaster (several hills) by marking peak and valley points and then by fitting (in height and slope) a cubic polynomial function to each consecutive pair of marked points.  Once the functions have been determined, students then calculate the thrill of the coaster.  A downloadable Maple worksheet with commands and explanation is provided.

F.  Project Assignment - Design the Most Thrilling Straight Stretch Coaster -  Students use the ideas from modules  A- E above to design a coaster (that satisfies all restrictions) with the maximum possible thrill.  Completion of this project requires ingenuity, creativity and extension/modification of many of the ideas and Maple commands presented in modules A-E.