All the various up, down and around designs all work because of gravity, inertia and friction. With their breathtaking elevation changes and speeds, spine-chilling roller coasters rides are the star attractions of amusement parks. This engineering curriculum aligns to Next Generation Science Standards ( NGSS). A pre-quiz, PowerPoint® presentation, spreadsheet calculations/graphs (with and without calculus), and student instructions/grading rubric are provided.
This activity and its associated lesson are designed for AP Calculus. 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. Students use Excel® to make these calculations and graph the designed path and velocities. 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. 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. The design must be efficient enough that the initial potential energy of the body is sufficient for it to complete the entire path. Project constraints students must consider include: initial cart velocity of zero (at the highest point), and final path end velocity of zero. 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. 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. 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.
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