In this video, the author presents a method to calculate shapes of roller coaster loops. At 13:20, three differential equations are presented to plot the shape of a loop providing a constant force G for an initial velocity v0:
∂θ∂s=G−gcos(θ)v20−2gy∂x∂s=cosθ∂y∂s=sinθ
where g is acceleration due to gravity, 9.80665 m/s2.
I would like to eliminate the need for the ∂θ∂s term. Do I integrate both sides of all these equations with respect to s and then substitute θ in for the x and y equations, or am I stuck with three equations?
Another problem is that the first equation has θ and y on the right side, so how would I proceed?
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