Designing a Frictional Roller Coaster With Math and Physics!
Students apply high school-level differential calculus and physics to the design of two-dimensional roller coasters in which the friction force is considered, as explained in the associated lesson. In a challenge the mirrors real-world engineering, the designed roller coaster paths must be made from at least five differentiable functions that are put together such that the resulting piecewise curving path is differentiable at all points. Once designed mathematically, teams build and test small-sized prototype models of the exact designs using foam pipe wrap insulation as the roller coaster track channel with marbles as the ride carts.
Project constraints students must consider include: initial cart velocity of zero (at the highest point), and final path end velocity of zero. The design must be efficient enough that the initial potential energy of the body is sufficient for it to complete the entire path. To achieve an efficient design, students use a formula obtained in the associated lesson—one that gives the velocity of a spherical body rolling on a curved path when friction is present. This equation considers the body’s energy lost due to friction, and is used to estimate the maximum height the marble may reach after rolling from a hill. Students use Excel® to make these calculations and graph the designed path and velocities. To conclude, teams summarize their procedures, designs, results, and theory-vs.-reality experiences in a slide or video presentation to their classmates, including their small-scale physical models. This activity and its associated lesson are designed for AP Calculus. A pre-quiz, PowerPoint® presentation, spreadsheet calculations/graphs (with and without calculus), and student instructions/grading rubric are provided.
Roller coasters are the star attractions in amusement parks. Since the first “steep hills and valleys” Russian Mountains to the thrilling multi-loops and turns provided by rollers coasters of today, all test the courage of the riders because of their extreme accelerations and velocities.Copyright (left) 2007 Jonrev, Wikimedia Commons (public domain); (right) 2006 Arsonwinter, Wikipedia (public domain) https://commons.wikimedia.org/wiki/File:Demon_Roller_Coaster.jpg https://en.wikipedia.org/wiki/File:Dorney_Park_Steel_Force_Thunderhawk.jpg
With their breathtaking elevation changes and speeds, spine-chilling roller coasters rides are the star attractions of amusement parks. All the various up, down and around designs all work because of gravity, inertia and friction. In this activity, by designing a simple roller coaster, students consider the same forces that professional engineers do when designing rides. Students apply mathematics, energy conservation principles and computation software to completely define the path and optimal dimensions for their designs. Then they use simple materials to build and test functional roller coaster models, which gives them the opportunity to address prototype construction problems and the unanticipated details that impact the expected design performance. Also like professional engineers, students find solutions to these problems.
After this activity, students should be able to:
- Create a design given certain requirements and constraints.
- Create, using parabolas, a piecewise differentiable function given specific vertices and determining tangency points.
- Estimate the velocity of a rolling body along a curved path, considering friction forces.
- Use computational software (such as Excel®) to evaluate and graph functions.
- Build a functional model from a mathematically generated design.
- Determine the possible causes why physical models might not behave exactly as expected in the theoretical, mathematically derived, “on-paper” design.