Roller Coaster Physics

picture of roller coasters

Photo credit: jekert gwapo

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.

roller coaster descending a height h

Source: Wikipedia via Chris Hagerman

Assuming no friction losses, when the center of mass of the roller coaster falls a vertical height h (from the initial hill) it will have a kinetic energy equal to the gravitational potential energy stored in the height h.

This can be expressed mathematically as follows.

Let W be the gravitational potential energy at the top of the hill.

Then,

equation for gravitational potential energy of roller coaster at height h

where m is the mass of the roller coaster, and g is the acceleration due to gravity, which equals 9.8 m/s2 on earth's surface.

The kinetic energy of the roller coaster is:

equation for kinetic energy of roller coaster

where v is the speed of the roller coaster.

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

conservation of energy for roller coaster

Thus,

conservation of energy for roller coaster 2

mass cancels out, and

velocity of roller coaster

This result is nice because it allows us to approximate the speed of the roller coaster knowing only the vertical height h that it fell (on any part of the track). Of course, due to friction losses the speed will be a bit less than this, but it is very useful nonetheless.


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. This type of acceleration can produce strong g-forces, which can either push you into your seat or make you feel like you're going to fly out of it.

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

roller coaster going around a loop

Photo credit: Joe Shlabotnik

where R is the radius of the loop.

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

centripetal acceleration experienced by the riders on a roller coaster going around a loop

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

roller coaster twisting around a track

Source: Wikipedia via BrandonR

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

centripetal acceleration experienced by the riders on a roller coaster 2

where w is the rate of twist (in radians/second) of the riders as they move along the track, and R is the radius of the twist at the location of the rider.


The acceleration experienced by riders on roller coasters can be quite high, as much as 3-6 g (which is 3-6 times the force of gravity).


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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