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Design of a Thrilling Roller Coaster

Patricia W. Hammer, Jessica A. King, and Steve Hammer

Texas Cyclone, Houston, TX

For the Instructor

Valencia, CA

In this project, you 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 a 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 you must design a straight stretch roller coaster that satisfies coaster restrictions (see the box below) regarding height, length, slope, and differentiability of coaster path and that has the maximum thrill, according to the following definition:

The thrill of a drop is the product of the angle of steepest descent in the drop (in radians) and the total vertical distance in the drop. The thrill of the coaster is the sum of the thrills of each drop.

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 path of the coaster must be modeled using differentiable functions.

Steel Dragon 2000
Nagashima, Japan

You must use Maple (version 8 or later) to complete the project. You 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.

To complete this project, you must complete Module A (next page) and as many of the Modules B through E as your instructor chooses to assign. You will then be ready to tackle Module F.

Trish Hammer is in the Department of Mathematics and Statistics at Hollins University. Jessica King is a 2004 graduate of Hollins University with a degree in Computer Science. Steve Hammer is in the Department of Mathematics at Virginia Western Community College.




All roller coaster images on these pages are used with permission of their respective copyright holders. Sources:

Published February, 2005


Patricia W. Hammer, Jessica A. King, and Steve Hammer, "Design of a Thrilling Roller Coaster," Convergence (February 2005)