PHYSICS+of++ROLLER+COASTERS

ROLLER COASTERS
Roller coasters are governed by and illustrate some of the most fundamental principles of physics. Almost 400 years ago, Galileo already knew many of the basic physical principles that underlie today's roller coasters. In particular, his "Dialogues Concerning Two New Sciences" (1638) contains discussions of free fall and descent along inclined planes. A roller coaster train going down a hill represents, in a sense, merely a complex case of a body descending an inclined plane. Newton developed the rest of the fundamental physics needed to understand roller coasters, by giving an improved understanding of forces. The first two of Newton's Laws in his "Principia" (1687) relate force and acceleration, which are key concepts in roller coaster physics. Newton was also one of the developers of the calculus, essential to analyzing falling bodies constrained on more complex paths than inclined planes. One physical concept useful to understanding the dynamics of roller coasters (though not essential, in the sense that one could perform all the calculations without it) is the concept of energy, which was developed by many physicists of the 19th century, though its roots extend to earlier physics. Albert Einstein, in "The Evolution of Physics", used a roller coaster as an example of energy conversion in a system. Finally, though the operation of roller coasters can be fully understood without any reference to the Theory of Relativity, coasters provide an illustration of Einstein's Principle of Equivalence, which underlies the General Theory of Relativity. A roller coaster rider is in an accelerated reference frame, which by the Principle of Equivalence, is physically equivalent to a gravitational field. This is why we measure the forces exerted by coasters in units of G's, where 1 G represents the force the rider experiences while sitting stationary in the earth's gravitational field. As the acceleration on the rider changes, the G forces will change as well.

Potential and Kinetic Energy
Fundamentally, a roller coaster is a very simple machine. The train is carried up to the top of the lift hill by the lift motor, and is from then on "powered" by gravity until it returns to the station (for the sake of simplicity, I will ignore unusual cases such as shuttle loops). This process is most easily thought of in terms of energy. There are many forms of energy, but for the roller coaster, two types are crucial: potential and kinetic. Kinetic energy is the easier of the two to understand--it is simply energy of motion. The faster a body moves, the more kinetic energy it has. Potential energy is more difficult to grasp. It can be thought of as stored energy. Consider when you lift a heavy object. To do this, you exert energy. This energy becomes available as potential energy, which can then become kinetic energy when you drop the object. Similarly, the lift motor of a roller coaster exerts energy to lift the train to the top of the lift hill, energy that will eventually become kinetic energy when the train drops. Lifting the train higher gives it more potential energy. This potential energy is converted to kinetic energy when the train drops. The further it drops, the more potential energy that gets converted to kinetic energy. In other words, the train picks up speed as it falls. After the train disengages from the lift chain, it receives no further source of energy--it operates entirely on the energy it got by being hauled up the chain lift. At the tops of hills, it has a lot of potential energy and only a little kinetic energy (it is high up, and moves slowly). At the bottoms of dips, it has a lot of kinetic energy and only a little potential energy (it is moving quickly, and is near the ground). Though it may sound as if we are just using complicated words to express obvious facts, analysis in terms of energy is very powerful and useful. Because (neglecting friction) the sum of the potential and kinetic energy is always constant, the speed of the train at any time depends only on how far it is below the highest point of the ride. Thus, we have a very simple way of determining how fast the train will go under ideal circumstances. This helps us understand why the //Steel Phantom// is a bit faster than Desperado. Though the largest drop of each is 68.5 meters, //Desperado's// big drop begins at the top of its lift hill, while //Steel Phantom's// big drop is its second, and begins below the height of the lift hill. The total vertical drop is greater for //Steel Phantom// than for the //Desperado//, even though it is not taken all at once. Since the potential energy difference depends only on how far the train falls vertically, not how it gets there, the //Phantom's// train will gain more kinetic energy. Another way of thinking about this is that //Phantom's// train is travelling faster at the beginning of its big drop, because it has already fallen somewhat below the height of the top of its lift. (This analysis does not account for friction and air resistance, which also play a role in the top speeds of these coasters. Unfortunately, these factors are much more difficult with which to deal. However, they can reasonably be left out of a first analysis. We will return to them below.) In theory, the train at the bottom of the first drop should have enough energy to get back up to the height of the lift hill. In practice, of course, this never happens, because some energy is lost to dissipative forces (such as friction and air resistance), which we shall discuss later. Also, if the coaster has mid-course brakes, the train loses energy to them, meaning that it can neither go as fast nor as high after the brakes as it would if they weren't there. At the end of the ride, of course, the remaining energy of the train is spent on the brake run.

Free Fall
A body that falls under no other influence than the force of gravity is undergoing free fall. If it starts from complete rest, it falls straight down, accelerating as it falls. If it starts with some horizontal motion, the path it takes will be a parabola which points downwards. The shape of a parabola is similar to (though not exactly the same as) the St. Louis arch. The ancient Greek geometers studied this shape extensively, but Galileo was the first to recognize its connection with falling bodies. A parabolic hill is a particularly special kind of coaster hill. When the train goes over such a hill, it, and its riders, briefly undergo free fall. In this case, the train may literally not be touching the track at all. Because, neglecting air resistance, all bodies fall at the same rate (another fact known to Galileo), the riders will fall in synchronization with the train. No part of the train will exert any force on the rider; the only force involved is gravity. In a sense, the rider is actually flying, because he or she is taking the same path as if there were no train or track there at all. In the 1920s, a coaster was built (The //Cannon Coaster// at Coney Island) based on this idea. It had a gap in the track where, theoretically, the train would take a parabolic trajectory, and land on the other side of the gap. However, the concept failed (fortunately without killing anyone). Though in theory, the train should have taken the same path each time across the gap, in practice, differences in conditions from run to run, such as wind speed and direction, meant that it would never take a completely consistent path, so that it wouldn't always land properly. The higher an object falls from, the faster it will be going when it reaches the ground. However, this is not a direct proportionality. An object dropped from twice the height does not go twice as fast. Since it is going faster on the second half of its drop, it has less time to accelerate to pick up additional speed. Thus, the top speed of the //Magnum XL-200// is less than twice as fast as the //Blue Streak's// (115 km/h versus 64 km/h), though its first drop is more than twice as high (59 meters versus 22 meters).