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Basic Roller Coaster Physics

Roller coasters work on the basic principle of energy conservation and conversion. What this means is that, technically, a roller coaster train has the exact same amount of energy throughout its cycle, but just in different forms.

The two main forms of energy are potential and kinetic. Potential energy is dependent on the mass of the train, its height above the ground, and the acceleration of gravity. This relationship means that the higher Superman's train is, the more potential energy it has. It also means that the more mass it has (more people!), the more potential energy it has, as well.

Kinetic energy is energy in motion and depends on the mass of the train and its velocity.

Using these concepts, Superman's train can then be described by gaining potential energy as it ascends the lift hill. Because, as stated, the amount of energy is conserved, this potential energy must be coming from somewhere. The answer lies in the lift motor. The power (and thus, work) delivered by the motor is converted to potential energy in the train. (Note: Because the train is moving up the lift, too, it has a small amount of kinetic energy as well.)

So once Superman's train reaches the 208-foot apex, it has a great amount of potential energy and a little bit of kinetic energy. As it rounds the top and begins to descend, the potential energy begins to convert to kinetic energy (because the height decreases), causing the increase in velocity. When the train reaches the bottom of the first drop, its kinetic energy is at its maximum and there is little or no potential energy (actually, getting more complicated, there would be a negative amount of potential energy due to the underground tunnel - the height would be negative).

This conversion between kinetic and potential energy continues throughout the cycle and is what allows Superman to complete the course.

But if the amount of energy in Superman's train is supposed to be constant throughout the cycle, then why is the train obviously slower and can only go over only small hills by the end? The answer is simple: Some of the energy is also converted into energy lost by friction and drag. Every joint that moves and every wheel that touches the track causes a loss of energy through friction. And because Superman isn't 100% aerodynamically sound (especially with 36 riders with hands up), some energy is lost through wind resistance (drag). This explains why each hill is successively smaller and Superman's average speed slowly decreases.

Kinetic Energy / Potential Energy / Energy lost from Friction & Drag

This explanation of energy provides answers as to how a 27,000-pound train can so easily climb a hill, but it does not explain why. This answer lies in the principle of intertia. Inertia is the tendancy of every object in space to continue in a straight-line path unless acted upon by a force. In the case of Superman, these forces include gravity and the track "pushing" on the wheels to force the train in a certain direction.

More Technically



E = 1/2mv2 + mgh + umg x cos(a) x d at any point along the circuit

    m = mass of the train, v = velocity of the train, g = acceleration of gravity,
    h = height above the ground, u = coefficient of friction, a = angle of the train,
    d = change in distance

    m = 27,000 lbs.
    g = 32.17 ft/s2
    u is dependent on the weather conditions, humidity, etc.

For example, assuming the train is traveling at 7 mph as it rounds the top of the first hill (reasonable) and that it does not lose any energy through friction on the first drop (unreasonable, but simple), the theoretical speed can be found at the bottom of the first hill.



Because the energy has to remain constant, setting these equal, we get

    1/2mv2 + mgh = 1/2mv2 + mgh + 0

First, we must convert the 7.00 mph to ft/s so that all units are consistent (the height is in feet and the acceleration of gravity - 32.17 - is in feet/second2).

    (7.00 mi/hr) x (5,280 ft/mi) x (1 hr/3600 s) = 10.267 ft/s

Now, we can plug the given values into the above equation to find v.

(1/2)(27,000)(10.267)2 + (27,000)(32.17)(221) = (1/2)(27,000)v2 + (27,000)(32.17)(0)

Factoring out the 27,000 lbs from each term and solving, we get:

    52.702 + 7,109.57 = 1/2v2

    7,162.272 = 1/2v2
    v2 = 14,324.54
    v = 119.69 ft/s

    (119.69 ft/s) x (1 mi/5280 ft) x (3600 s/hr) = 81.6 mph

81.6 mph is clearly not an unreasonable answer, as this is assuming an empty train (27,000 lbs.) with no friction. The actual speed of an empty train hovers around 77mph (friction accounts for the difference), but the train can achieve this higher speed with people in it (making the actual weight as high as 34,000 pounds).

The Forces on Riders

There are three main types of forces felt by riders while on Superman. They are vertical G's, lateral G's, and longitudinal G's. As described on the Forces page, vertical G's are what cause riders to feel as if they are coming out of their seats (or, just the opposite - being pushed in), lateral G's cause riders to move side to side, and longitudinal G's cause riders to move forward or back in their seats.

The root of these forces is the basic concept of intertia. As stated, an object in motion tends to continue in a straight-line path unless an outside force is applied. In addition to this helping the train to move along the track, it also affects the riders in the train.

As for the vertical G's, these are most noticeable at the tops and bottoms of in-line hills (when the other two types of forces are minimal). As the train rounds the top of the second hill, for example, the train is being pulled over the top by the track, however, the riders' bodies want to continue upward (or at least in a parabolic projectile motion path). This is the cause of the out-of-seat, air-time feeling. The opposite effect is at the bottom of a hill when the train begins moving upward and the riders' bodies want to continue downward or straight ahead. The force of the train pushing up on the body is what makes riders feel heavier.

The same goes for the other two forces: Lateral and longitudinal. In the case of the former, the forces are felt as the train moves to the left or right and the body wants to continue straight ahead. And in the case of longitudinal forces, any change in speed will cause the body to move forward or backward.


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