Amusement Park Physics

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A new era in theme parks and roller coaster design began in 1955 when Disneyland ushered in the new era of amusement park design. Disneyland broke the mold in roller coaster design by straying from the typical norm of wooden roller coasters; thus, the steel tubular roller coaster was born. Disneyland’s Matterhorn was a steel tubular roller coaster with loops and corkscrews, which had never been seen before with the wooden coasters. In addition to the new steel tube roller coaster, the new coaster design also proved to be the most stable, allowing for wilder designs. The first successful inverted roller coaster opened up in 1992, and now it is not uncommon to find passengers of various roller coasters with their feet dangling above or below them as they circumnavigate the track. In 1997 Six Flags Magic Mountain opened a roller coaster, that just a few year previous would have been considered impossible. The Scream Machine is 415 feet tall and takes willing riders on an adrenaline rush using speeds of 100 miles per hour. Technology working with the laws of physics continues to push the limits of imagination and design.

Many people do not realize exactly how a roller coaster works. What you may not realize when you are cruising down the track at over 60 miles per hour, is that the roller coaster does not have a motor or engine. At the beginning of the ride the car is pulled to the top of the first hill where it comes to a momentary halt. At this point its potential energy is at a maximum and the kinetic energy is at a minimum. As the car falls down the hill it is losing potential energy and is gaining kinetic energy. It is this kinetic energy that keeps the car going throughout the remainder of the ride. The conversion of potential energy to kinetic energy is what drives the roller coaster, and all of the kinetic energy you need for the ride is present once the coaster descends the first hill. Once the car is in motion, different types of wheels keep the ride running smooth. Various running wheels help guide the coaster around the track. Friction wheels control lateral motion. A final set of wheels keeps the coaster on the track even if the coaster is inverted. Compressed air brakes are used to stop the coaster as it comes to an end.

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"Amusement Park Physics." 16 Oct 2017
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A roller coaster ride begins with an engine hauling a train of cars up to the top of a steep grade and releasing them. From this point on the train is powered by gravity alone and the ride can be analyzed by using the fact that as the train drops in elevation its potential energy is converted into kinetic energy.

It is not too hard to derive a formula for the time t required for a ride along any curve y = y (x) that the roller coaster track takes. The result is the following formula:

Once the curve y (x) for the roller coaster track is given, it and its derivative y'(x) can be substituted into this formula and the integration can be carried out.
At the point x in the picture the potential energy lost is mgy so the kinetic energy at that point is given by:

Solving for v gives:

The following notation y(x) and v(x) has been written instead of just y and v to emphasize the fact that the elevation drop and hence the velocity both depend on the horizontal coordinate x.) This is the basic formula which can now be manipulatedand solved.

First, the velocity v is measured along the curve so we must rewrite it in terms of its horizontal and vertical components. To do this let:
• s represent the accumulated distance along the curve,
• ds represent a small incremental distance along the curve, and
• dx and dy represent the horizontal and vertical components of ds
• ds/dt represents the velocity along the curve,
• dx/dt represents the x component of the velocity,
• dx and dy are given by Pythagoras' theorem, and
• the chain rule gives:

Substituting this into our basic formula gives:

At this point the curve y(x) for the roller coaster track must be specified. Then it and its derivative y'(x) can be substituted into this formula and this is seen to be a differential equation whose solution is some function x(t).

Potential Energy in a Gravitational Field

Potential energy is the energy that an object possesses by virtue of its location. If gravity is the only force present then the potential energy is given by the formula:
potential energy = m g h

where m is the mass of the object, g is the acceleration due to gravity (the gravitational constant), namely 9.81 m/s2, and h is the height of the object. If the object is raised then the object gains this much potential energy; if the object is lowered then it loses this much potential energy.

In the SI (metric) system of units:
• m is expressed in kg (kilograms),
• g is expressed in m/s2 (meters per second squared),
• h is expressed in m (meters),
• the energy is expressed in J (joules).

Potential energy (in fact any type of energy) can be converted into other forms of energy. For example if an object is dropped then it speeds up (gains kinetic energy) as at falls (loses potential energy).

Kinetic energy is the energy that an object has by virtue of its motion. It is given by the formula:
kinetic energy = ½ m v2
where m is the mass of the object and v is its velocity.

In the SI (metric) system of units:
• m is expressed in kg (kilograms),
• v is expressed in m/s (meters per second),
• the energy is expressed in J (joules).

Note that if the velocity of an object is doubled, its kinetic energy is quadrupled.

Kinetic energy can also can be converted into other forms of energy. For example on mountain roads runaway lanes are often provided for trucks that lose their brakes while going down long hills. The runaway lane takes the truck back uphill and allows the truck to coast to a stop as its kinetic energy is converted to potential energy.

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