Modeling a Simple Roller Coaster

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This Demonstration is an animation of a simple roller coaster. The coaster track is restricted to a vertical plane 100 meters long and 50 meters high. The car slides on the track without friction and is subject to gravity only.


Create a custom track by moving, creating, or deleting the locators in the upper plot or select one of the example tracks.

Keep the track inside the plot and the bounding box. The track will be capped and the animation is not correct outside this range.


Contributed by: Erik Mahieu (May 2011)
Open content licensed under CC BY-NC-SA



In the absence of friction, if represents the shape of the track, then the time for the car to move from horizontal positions to is: . In this Demonstration, this integration is approximated numerically with 100 subdivisions of the track and by using interpolation. This speeds up the calculation during the animation while sacrificing some accuracy.

Snapshot 1: the fastest ride possible along any track between the points and is the brachistochrone of duration or approximately 8 seconds

Snapshot 2: the time for the circular track ride is or 8.37 seconds

Snapshots 3–6: all other tracks have rides longer than 8 seconds

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