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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