The main energy transfers that happen as a car travels along the track from the start of the ride to the end:

[1]The rollercoaster car gains gravitational potential energy (GPE) as it travels to the top. Once over the top, the car gains speed as GPE is transferred to kinetic energy (KE). As it travels to the top of another loop, KE is transferred to GPE. Not all the energy is transferred to or from GPE – some is transferred to the surroundings as heat and sound. All moving objects have kinetic energy, KE. The kinetic energy an object has depends on the mass and speed. If the mass doubles, the KE doubles and if the speed doubles, the KE quadruples. Normally energy is lost through sound and heat (friction, air resistance).

How the heights of the hills are designed to allow an empty car to reach the end of the ride:

[2]The hills are designed so that it is low enough that the momentum of the car from the previous drop carries it up and over the hill. This is why the hills are usually lower towards the end of the ride, because the car has lost momentum due to friction and air resistance. Mainly the consecutive hill must be lower as it will not have enough energy because some of it is lost and sound and heat. Therefore, if the car was to reach the end of the ride, the height of the hills must be lower each consecutive time.

How the energy transfers determine the heights of the hills:

[4] After the roller coaster is drops from the first hill it does two things with its energy. First, it begins to transform that energy from one form to another--from gravitational potential energy to kinetic energy and from kinetic energy to gravitational potential energy, back and forth. Second, it begins to transfer some of its energy to its environment, mostly in the form of heat and sound. Each time the roller coaster goes downhill, its gravitational potential energy decreases and its...

...RollerCoasters
The main energy transfers that happens as a “car” travels along the track from the start of the ride to the end.
1. The main energy transfers are between gravitational potential energy (GPE) and kinetic energy (KE), and the eventual decrease of mechanical energy as it transforms into thermal energy. Rollercoasters often start as a chain and motor exercises a force on the car to lift it up to the top of a very tall hill. At this height, GPE is at its highest, as we can see through the formula:
GPE = mass x gravitational field strength x height (for all physics in relation to Earth, take g to be 10 m/s2 or 10 N/kg)
We can see through this formula that as the height increases, so does the GPE, which will then be converted into KE, or kinetic energy. This is the energy that takes place as the “car” is falling down the hill. This is calculated through the formula:
KE = 0.5 x mass x speed
This means that the kinetic energy increases as the speed increases, and vice versa. Therefore, this means the higher the kinetic energy, the faster the “car”. We can actually be extremely specific in terms of this relationship. We know that as the mass doubles, the KE doubles, but as the speed doubles, the KE quadruples. This becomes important when analysing this formula:
KE = GPE/0.5mv2 = mgh
2. A rollercoaster ride is a thrilling experience which involves a...

... The cars subsequently gain kinetic energy. Kinetic energy - the energy of motion - is dependent upon the mass of the object and the speed of the object. The train of coaster cars speeds up as they lose height. Thus, their original potential energy (due to their large height) is transformed into kinetic energy (revealed by their high speeds). As the ride continues, the train of cars are continuously losing and gaining height. Each gain in height corresponds to the loss of speed as kinetic energy (due to speed) is transformed into potential energy (due to height). Each loss in height corresponds to a gain of speed as potential energy (due to height) is transformed into kinetic energy (due to speed)
http://www.physicsclassroom.com/mmedia/energy/ce.cfm (7/10)
A rollercoaster moves in the same way a marble would roll down a slanted surface. The marble rolls because it has Gravitational Potential Energy. Potential Energy is gathered by an object as it moves upwards, or away from, the earth. With a rollercoaster, this is acheived by pulling the train up a lift hill to the coaster's highest point. As it moves higher, it has more potential to fall to earth, increasing its Kinetic Energy. Kinetic Energy is gathered as an object falls. There's a transfer of Potential Energy to Kinetic Energy as the rollercoaster train leaves the top of the lift hill and enters the first drop....

... Physics behind rollercoasters
Energy can be converted from one from to another. When the car is still, the energy which is acting on it is GPE (gravitational potential energy). The car starts to accelerate towards the peak. The energy is converted from GPE to Kinetic energy. The car is at the peak. The energy transfers from Kinetic energy to GPE. The car starts to go down. The energy transfers again from GPE to Kinetic energy. There is no kinetic energy when the car is still at the bottom, as the energy has converted to GPE. The higher the lift, the more potential energy gained by the car. Not all the energy is converted or transferred as GPE and kinetic energy. Some of the energy is transferred as thermal energy and sound energy. Kinetic energy - the energy of motion - is dependent upon the mass of the object and the speed of the object. The train of coaster cars speeds up as they lose height. When a rollercoaster travels at a constant speed on a circular part of the track, it is accelerating – its velocity is changing because it is changing direction. There is a resultant force making the train move in a circle. The resultant force and the acceleration are directed towards the centre of the circle.
Mass- if the mass doubles, the KE doubles
Speed- if the speed doubles the KE quadruples
The equation above shows how the speed and velocity are calculated
Potential Energy is gathered by an object as it...

...geometry and calculus all take a huge role helping out in forming these models.
Rollercoasters first originated in the 16th and 17th centuries by the Russians. They would create sleds made of ice or wood and slide down slopes. The French were amused with this pass time that they actually took the idea back with them home. On HowStuffWorks.com it states, “The most widespread account is that a few entrepreneurial Frenchmen imported the ice slide idea to France.” This was the first time that people were amused with any thought or idea of sliding down some kind of slope and feeling a rush that someday would turn out to be one of the favorite pastimes to present day. It was not until years later when La Marcus Thompson, an inventor and designer created the first rollercoaster in Coney Island, New York City. In Wired.com it states, “…hurtled passengers down an undulating 600-foot-long track at speeds of up to a blistering 6 mph would hardly be recognizable to riders of modern-day rollercoasters.” The coaster was a 50 foot platform and the passengers would sit sideways. This was the beginning of the rollercoaster.
How does math come into play then? Math is needed to build a rollercoaster because everything has to be precise. The first drop is one of the most important because it will determine the speed for the rest of the ride. The height and...

...Amusement Park Physics
For: Mr Day
Due: 01/04/08
By Rhys Webb
Synopsis 1
Introduction 2
Brief history of RollerCoasters 2
Physics of rollercoasters 2
Rollercoaster Design 3
Analysis of RollerCoaster 4
1st Slope 4
2nd Slope 4
1st Dip 5
3rd slope 5
Loop 5
Conclusion 5
Bibliography 6
Appendix 6
Synopsis
The context of this report is to design and analyse a rollercoaster within the parameters of: a maximum “g” force of “4g’s”, a length of 40 to 100 seconds and has to be constructed of metal rather than wooden trestles. This report also requires a qualitative and quantitative explanation of the theory and figures behind the analysis.
Introduction
In this report there is a qualitative and quantitative explanation of the physics of a rollercoaster and well as the figures which are retrieved via mathematical analysis of a rollercoaster.
Brief history of RollerCoasters
The first basic rollercoasters were first created in Russia in the 1780’s where a large wooden ramp would be constructed in winter and as the ice covered it people would ride sleds down it, these creations were called ‘Russian Mountains’. In 1804 the innovative idea was...

...Samantha Robinson
Math105
Math and RollerCoasters
It’s not something that most people think about when they’re standing in line for an hour waiting for two minutes of thrill, but math has a lot to do with building a rollercoaster. The engineer needs to know how long and how high the ride needs to be before it can begin to be put together. These measurements have to be precise because the first drop has to give it enough speed to carry it through all the features of the ride. They would also have to take into consideration whether it will be made out of wood or steel because a wood coaster can not be as high as a steel one. The weight of the cars plays a part in the structure since it can’t be too heavy or this will interfere with the speed. It will also cause it to fall down the steep sloop it travels to get to the first drop. To name a few, geometry, algebra and trigonometry are used when a rollercoaster is being designed.
There are various types of rides and they all have to be built differently.
The corkscrew is shaped like a twisted spiral. Geometry definitely plays a big part in the creation of this ride because the track rails have to be equal so the wheels can stay in sync.
When a launch coaster is being put together the engineer needs to calculate the amount of stress that can be put on the launching gear. This again, is where the...

...er 3) Physics of rollercoasters
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Simply speaking, a rollercoaster is a machine that uses gravity and inertia to send a train of cars along a winding track.[1] This combination of gravity and inertia, along with G-forces and centripetal acceleration give the body certain sensations as the coaster moves up, down, and around the track. The forces experienced by the rider are constantly changing, leading to feelings of joy in some riders and nausea in others. The basic principles of rollercoaster mechanics have been known since 1865,[citation needed] and since then rollercoasters have become a popular diversion.
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centripetal acceleration
Centripetal acceleration is not a true force, but rather the result of an object’s inertia, or resistance to change in direction, as the object moves in a circular path. The "force" points toward the center of the circle, but a rollercoaster rider would feel centripetal acceleration as a force pushing them toward the outer edge of the car. The following equation expresses centripetal acceleration:
where ar is centripetal acceleration, v is velocity and r is the radius of the circular path. This shows that two rollercoaster cars entering two loops of different size at the same speed will experience different...

...The rollercoaster is a popular amusement ride developed for amusement parks and modern theme parks. Most rollercoasters consist of some basic parts: a chain lift, a catapult launch-lift (in newer coasters), and some type of brake mechanism. The first rollercoaster originated in the early 80’s in Russia, it was built under the orders of a Russian empress named Catherine the Great in the Garden of Oranienbaum a Russian royal residence located in Saint Petersburg.
Rollercoasters work by using gravity to move along the track, most think they have an engine inside that pushes them along the track like an automobile. The first part of a rollercoaster is always a big hill; the biggest hill is always the highest point. The purpose of this is to gain two very important things gravitational potential energy, and kinetic energy. The reason these two things are important is because over the course of the coaster ride those two things propel the coaster, by exchanging for one another and sometimes combining. They can be exchanged for one another because at certain points on the ride the rollercoaster may just have potential energy (like at the top of the first hill), or just kinetic energy (like at the lowest points) or maybe a combination of both kinetic and potential...