One of the challenges that roller coaster designers face is controlling the forces their passengers experience as they take the ride.

The forces in a vertical plane are simple to state. They are the sum of the gravitational force pulling down and any centrifugal force that depends on radius of curvature and acts normally to the track.

The easiest track to design is circular but this causes a problem, “In practice this leads to unpleasantly large time variation of the normal force,” say Arne Nordmark and Hanno Essat the KTH Royal Institute of Technology in Stockholm. So designers are forced to rely on other more complex shapes, such as the clothoid.

So a reasonable question to ask is what shape should the tracks be to deliver a constant smooth normal force to a passenger taking the ride. Such a roller coaster would offer the perfect smooth ride.

(A curious feature of Nordmark and Essat’s analysis of this problem is that they say it is related mathematically to Kepler’s problem of planetary motion. Who’d have guessed.)

The answer, say the Swedes, is the shapes in the picture above. These do not yet appear to have a name–Nordmark and Essat call them Case 1 trajectories. Readers of the arXiv Blog will surely have better ideas. Suggestions in the comments section please.

Ref: arxiv.org/abs/1007.1394: The Comfortable Roller Coaster – On The Shape Of Tracks With Constant Normal Force