Source: http://www.flickr.com/photos/jekert/3089042791

One can best gain an appreciation of roller coaster physics by riding on a roller coaster and experiencing the thrill of the ride. There are many variations on roller coaster design. But needless to say, they all involve going around loops, bends, and twists at high speed.

The typical roller coaster works by gravity. There are no motors used to power it during the ride. Starting from rest, it simply descends down a steep hill, and converts the (stored) gravitational potential energy into kinetic energy, by gaining speed. A small amount of the energy is lost due to friction, which is why it's impossible for a roller coaster to return to its original height after the ride is over.

The roller coaster uses a motorized lift system to return to its original position at the top of the initial hill, ready for the next ride.

The figure below illustrates the concept.

Source: http://en.wikipedia.org/wiki/Flying_roller_coaster. Author: http://en.wikipedia.org/wiki/File:GL-X-Flight.jpg

Assuming no friction losses, when the center of mass of the roller coaster falls a vertical height

This can be expressed mathematically as follows.

Let

Then,

where

The kinetic energy of the roller coaster is:

where

If we assume no friction losses, then energy is conserved. Therefore,

Thus,

mass cancels out, and

This result is nice because it allows us to approximate the speed of the roller coaster knowing only the vertical height

Another important aspect of roller coaster physics is the acceleration the riders experience. The main type of acceleration on a roller coaster is

Centripetal acceleration occurs mainly when the roller coaster is traveling at high speed around a loop, as illustrated in the figure below.

Source: http://www.flickr.com/photos/joeshlabotnik/2479908266

where

The centripetal acceleration experienced by the riders going around the loop is:

Centripetal acceleration can also occur when the riders twist around a track, as illustrated in the figure below.

Source: http://en.wikipedia.org/wiki/Roller_coaster_elements. Author: http://en.wikipedia.org/wiki/User:BrandonR

The centripetal acceleration experienced by the riders twisting around the track is:

where

The acceleration experienced by riders on roller coasters can be quite high, as much as 3-6

In summary, the physics of roller coasters (in general) is a combination of gravitational potential energy converted into kinetic energy (high speed), and using this speed to create centripetal acceleration around different portions of the track.

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