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Roller Coasters: Mathematical Simulation Research Paper

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Updated: Oct 8th, 2021

Popularity of Roller Coasters

Historically, the popularity of the Roller Coasters was reasoned by the great fun, which people got from the velocity and seeming danger of the rollers. With the development of technologies, the safety level increased with the velocity and excitement. Actually, the Great depression caused the decline of the popularity, but Today roller coasters have regained popularity and people are building steeper, faster and more thrilling roller coasters.

The earliest RC in the USA

The first roller coaster in the United States was made by L. A. Thompson in Coney Island, New York, in June, 1884. Thompson’s firm appears to have controlled the roller coaster sphere, making at least twenty Switchback Railways in the U.S. from 1884 to 1888, and a far greater number of Scenic Railways starting in 1887.

The second RC was built in Pennsylvania in 1886 by Thompson’s company.

The next relevant patent is that of Richard Knudsen of Brooklyn (1889, Improvement in Inclined-Plane Railways). The model is similar to Thompson’s, but consists in this case of a symmetrical pair of down-tracks with similar lifting machines at both ends. The cars are lifted vertically at each end applying a pulley constrruction.

The earliest Roller Coasters were made of wood. As for the contemporary RCs, they are mainly of steel and carbon-filled plastic.

The Evolution of Roller Coasters

The roller coasters evolved from wooden rails and rollers and simple hills to steel rollers, and rails with loops, reverses and upside-down turns.

Five Theme Parks

Disney’s California Adventure Park is a theme park in Anaheim, California. The roller coaster is on the 1st place in popularity rate (The California Screamin’ roller coaster.)

The 2nd is the Legoland park to open, located in Carlsbad, California. IT has three roller coasters (Mini Dino Coaster, Dragon Roller Coaster and Wild Mouse-style roller coaster called the Technic Test Track)

Adventuredome indoor amusement park located in Winchester, Nevada on the Las Vegas Strip. It has includes the Canyon Blaster roller coaster. It features back-to-back vertical loops and corkscrews, and ends with a helix inside the mountain that takes up a large portion of the park. It is billed as the world’s largest indoor double-loop, double-corkscrew coaster.

The Great Escape & Splashwater Kingdom: A Six Flags Park. An amusement park and water park located in Queensbury, New York. Actually, Charley Wood, the owner of Great Escape Fun Park and Fantasy Island in Grand Island, New York successfully bid for The Comet and it sat in storage for a few years in Fantasy Island before making its way to the park in Queensbury, NY and reopening in 1993. Roller coaster enthusiasts recognize it as one of the best wooden roller coasters in North America.

Safety considerations

Among some of the most important things that designers must pay attention to when designing a roller coaster are the acceleration, the rate of change of acceleration “jerk,” and the rate of change of jerk — “snap.” The acceleration is extremely important in determining if a ride will be exciting and sufficiently scary without being uncomfortable. “Jerk” and “snap” measure the forces present upon various parts of the coaster at each point in its movement, and are pivotal in roller coaster safety.

Maintaining reasonable levels of “jerk” and “snap” ensure that neither the users nor the building framework are subjected to abruptly large forces, which could cause bodily harm or structural collapse.

Roller Coaster Length Speed Height Time Acceleration
Steel Dragon 2000 8,133 feet 95 mph 306 feet 59 sec 1.4 m/s2
California Screaming 6,072 feet 92 mph 300 feet 45 sec 1.8 m/s2
Desperado 5,843 feet 85 mph 255 feet 48 sec 1.7 m/s2
Mamba 5,600 feet 85 mph 225 feet 46 sec 1.6 m/s2
Steel Force 5,600 feet 82 mph 221 feet 47 sec 1.5 m/s2

Acceleration role

The thrill of roller coasters is not because of their speed, but rather because of their accelerations and to the senses of weightlessness and weightiness which they offer. Roller coasters thrill people due to their capability to accelerate downward one moment and uphill the next; leftwards one instance and rightwards the next. Roller coasters are about hastening; that is what makes them exciting.


This model appears to be the most safe and full of fun.

The first hill is 80 meter in height, which gives the maximum potential energy. The shape of the first hill provides thrilling, but safe and gentle moving forward, and leads to the low exit path, which helps to keep the velocity. The remaining speed permits to overcome the second (70 meters height) heel, and then go through the loop successfully. The elliptical shaped loop will be fun and will avoid the dangerous effects of a circular loop, which can’t yet be safely done.

The most unsafe model will be the following.

Due to the shape of the first hill coaster will follow the curve of a projectile, leave the track and become airborne part way down, bounce, and then crash at the bottom of the hill. The exit path needs to be slight, but not angled, and the loop is necessarily elyptical.

The safest, but least entertaining model:

Entertaining model

Low hills and no loop will just make no fun and thrill.

Distance, Rate of Speed, and Time Duration

Roller coater Distance Max Speed Duration
Expedition GeForce (Holiday Park) 4003 feet 74 mph 2.15
Voyage (Holiday World) 6442 feet 67 mph 2.45
Superman – Ride of Steel (Six Flags New England) 5400 feet 77 mph 2.30
El Toro (Six Flags Great Adventure) 4440 feet 70 mph 1.40
X2 (Six Flags Magic Mountain) 3610 feet 76 mph 1.55
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IvyPanda. (2021) 'Roller Coasters: Mathematical Simulation'. 8 October.

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