The following is an article by David Sandborg.
It originally appeared on Jim Serio's "World of
Coasters" (WoC) website. Thanks to both David
and Jim for permission to reproduce it here.
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THE FOLLOWING MAY NOT BE REPRODUCED
WITHOUT PERMISSION FROM THE AUTHOR ©
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.
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.
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).
Most of the time, a roller coaster represents constrained fall. That is, the track constrains the train's path, so that it does not take the same trajectory as it would if the track were not there. Galileo's inclined planes represent the simplest examples of constrained fall. Unlike a parabolic fall, the inclined plane provides partial support for an object falling down it, preventing it from falling as fast as it ordinarily would.
The basic rule of falling down an inclined plane is that the falling body will accelerate faster the steeper the plane is. If the plane is completely vertical, the body will be in free fall, which represents the fastest gravitational acceleration possible. If the plane is completely horizontal, the body will not fall at all. This principle applies to coasters, even though the track on drops is usually not straight, like an inclined plane. The steeper a drop, the faster the train accelerates down it. Thus, the train accelerates very quickly down the steep first drop of Magnum. On the other hand, the train accelerates very slowly down the shallow drop into the helix of the Beast (even if we discount brakes).
However, if a steep and a shallow drop are of the same height, the train will eventually reach the same speed at the bottom, if we ignore friction. Though the train accelerates more slowly on a shallow drop, it has a longer time in which to accelerate. The factors balance out exactly. This makes sense when we think of energy: The potential energy difference from the top to the bottom of a drop depends only on its height, not on its steepness.
An important concept when it comes to understanding coaster physics is the concept of G forces. Ironically, despite its name, gravity isn't a G force, but G forces are measured in terms of what you feel when you are sitting still in the earth's gravitational field. When in that state, you are in a 1-G environment.
Consider what you really experience in 1 G. The earth is pulling you down towards its center. However, this is also happening when you are falling, and experiencing less than 1 G. Therefore, there must be something else that accounts for the G force. That something else is the force of the seat pushing back up on you. If there wasn't such a force, you'd fall through the seat. When you're sitting still, the seat must exert the same amount of force as the earth, directed oppositely, to keep you sitting still.
Now consider when you're in free fall. In that case, the seat isn't supporting you at all; that's why you're falling. Thus, the seat exerts no force on you, and you are said to be experiencing 0 Gs. In intermediate cases, such as when you are going down a straight incline, the seat supports you only partially (which is why you accelerate downwards--the seat doesn't completely balance the force of gravity), and you experience somewhere between 0 and 1 G.
G forces can also be thought of in terms of weight. If you sat on a scale, it would register your ordinary weight when you were sitting still, in 1 G. If you were in free fall, under 0 Gs, it would not register you as having any weight at all. In-between, it would register some fraction of your weight; depending on the fractional Gs you experienced.
So far, we have considered G forces between 0 and 1. You can also experience G forces greater than 1. This generally happens at the bottom of drops. There, the seat must not only prevent you from falling, it must also begin to divert your path upwards again, so it must exert a force greater than it would if you were sitting still. Thus, you experience greater than 1 G. In this case, you feel yourself being pushed down into the seat. What's really happening, though, is that the seat is pushing up on you...
Suppose the top of a hill is curved more sharply than a parabola. The train is locked to the track, and will follow this path. However, the rider will tend to fall along a parabolic path, and will thus rise above the seat. If this situation is prolonged, the rider eventually will hit the lapbar, which will then exert a downward force, to keep the rider in the seat. Since G forces are measured positively when the train exerts an upward force on the rider, they are measured negatively when the force is downward. Most coaster enthusiasts value these so-called "negative Gs" very highly.
Newton's laws contain the principle of inertia; that a moving body, if unaffected by any forces, will travel in a straight line. A coaster train is affected by forces along its run (the force of gravity, as well as the supporting force of the track), but if it is travelling along a straight-away, none of the forces will be directed towards either side. The riders, too, will not experience any forces to either side. On the other hand, if the train hits a curve, it will tend to want to go forward. The track has to exert a sideways force on the train to divert it from its path. The train, in turn, exerts a force on the riders. As they continue to try to go straight, they get pinned to the side of the car. Though they feel themselves being forced toward the outside of the curve (an effect commonly referred to as centrifugal force), the force that is exerted on them is actually towards the inside (centripetal force), because that is the direction in which they are turning. As with forces directed vertically, lateral forces can be measured in terms of Gs. A 1-G lateral force would be equivalent to your lying on your side.
Several factors affect the strength of lateral G forces: the speed of the train, the tightness of the curve, and the amount of banking. The faster the train goes through the curve, the greater the force required to keep it on the track. Similarly, the tighter the curve, the more force is exerted on the train. For example, the train going through the helix section of the Kennywood Thunderbolt is moving quite fast, because it has just come off a substantial drop. This explains its legendary lateral Gs. On some coasters, the curve may also contain a drop. In this case, the train speeds up through the curve, and the G forces get stronger. The Riverview Bobs contained two dropping turns of this sort with little banking. Pictures show the riders pinned to the outside of the car, as one would expect.
Banking a turn is a way to convert lateral G forces into positive Gs. If the track tilts towards the inside of the turn, then the train will tilt so that the floor, rather than the side, of the car is exerting some of the forces on the rider. Thus, some of the G forces experienced are now positive Gs. Note that the total force on the rider with respect to the ground is the same whether the curve is banked or unbanked, but the force relative to the train is different. Also note that if the track were tilted to the outside of the turn, the lapbar would exert some of the force keeping the rider in the train, that is, we could generate negative Gs through a turn!
As a rule, the higher the banking, the less lateral and the more positive are the Gs. Thus, the turn at the bottom of the big drop of the Steel Phantom is heavily banked, so that though the train is going very fast, and the turn is fairly tight, most of the force experienced is positive G force. On the other hand, the final helix is taken somewhat more slowly, but there is much less banking, so that the riders experience some lateral G forces.
When going through a turn, the force of gravity is still pulling the rider downwards. If the lateral forces are relatively weak and the turn is banked very much, this force becomes significant. There is a certain point of banking where the lateral force turns into positive G force. If the turn is banked beyond this, then the rider is pulled towards the inside of the train rather than the outside, and the turn is overbanked.
Many steel coasters nowadays go upside down. As you might imagine, physics plays a crucial role in this element as well. Let us consider a vertical loop. The basic idea is similar to what happens in a turn: Because the coaster train tends to go in a straight line, and the track impedes this, there is an apparent outward force. Thus, in a loop, positive Gs are generated, and the train doesn't fall off the track, nor the rider out of the car. Since a positive G force means that the seat exerts a force on you away from the floor of the train (though you feel like you are being pushed in your seat), the force that is exerted on you by your seat at the top of the loop is actually towards the ground (in addition to the downward pull of gravity)!
Inverting coasters all have wheels underneath the track, and most have over-the-shoulder restraints, but theoretically, these should not be necessary. In fact, an early inverting coaster, the Loop-the-Loop at Coney Island had neither of these things, apparently without mishap. However, wheels under the track are still prudent, because should the train somehow lose speed at the top of the loop, physics would no longer guarantee that it stays on the track. Even in this case, lap bars should be sufficient to hold a passenger in; shoulder harnesses were apparently designed to give a psychological sense of safety.
As with a turn, the forces exerted by a loop are determined by the speed of the train and the size of the loop. The faster the train or the tighter the loop, the more positive Gs are exerted. This principle accounts for the fact that vertical loops tend not to be circular, but more elliptical. Actually, most are a special shape called a clothoid loop (spellings seem to vary on this word; Robert Cartmell uses "Klothoide" in his "Great American Scream Machine"). When the train enters the loop, it is going relatively fast, and so the loop at this point is not very tight. At the top of the loop, the train is moving much more slowly, so the loop is tighter there. In this way, the G forces are somewhat better balanced than in a circular loop.
So far, we have neglected friction, one of the most important factors in coaster physics, because it complicates the picture somewhat. Of course, coaster designers can't afford to ignore it. As the train goes over the track, it loses energy to friction (from the track and wheel bearings) and air resistance. As stated above, without friction, the sum of potential and kinetic energy will remain constant after the train leaves the lift. Taking friction into account, this sum will continually decrease throughout the ride. So later in the ride, the train can't climb as high as it could in the beginning. To climb a high hill may require more energy than the train has left. Furthermore, at the bottom of hills, the train will tend to go slower at the end of the ride than it did at the beginning, because to go fast also means having a lot of energy. This only represents a very rough analysis. The precise computation of these factors is very difficult.
If a coaster feels faster at the end of a ride than at the beginning, this is partially an illusion. A well-designed coaster can still exert more extreme forces at the end of the ride than at the beginning (for instance, by making the turns tighter or less banked to make up for the loss of speed). Furthermore, the train's average speed can be greater near the end of a ride than at the beginning, because it may be going over a series of small speed bumps, rather than high hills over which it moves more slowly. However, the train reaches its top speed at the bottom of the first drop. (This is not strictly true. The last drop on Kennywood's Thunderbolt is deeper than earlier drops, and so the train reaches top speed there. In this, as in many other ways, the Thunderbolt is unusual.)
Let us return briefly to the comparison between the top speeds of the Steel Phantom and Desperado. Desperado's big drop is its first, so that the train has only traversed a relatively small stretch of track by the time it reaches the bottom of the drop. Steel Phantom's big drop is its second, so the train has gone over a somewhat longer stretch of track, and has been subject to more dissipative forces. Thus, The Phantom will not be as fast as if it went from the top of its lift hill to the bottom of the second drop all at once, rather than over two drops, and the speeds of the two coasters will be more similar than the potential energy analysis alone would predict.
Another complication we have not yet considered is that a coaster train is long. Different forces can operate on the front and the back of the train. Often, the ride in the front is distinctly different from the back.
Think of a train going over the top of the lift hill. There is a certain point at which it is precisely balanced on the crest of the hill. Before this point, it is driven by the chain. After this, gravity takes over, and it starts accelerating. Before this balance point, the front of the train has already started to go slowly over the top, at the speed of the lift chain. By the time the back gets over the top, the train is already moving faster. Thus, the front seat rider often has a sensation of dangling over the drop, while the back seat rider feels yanked over it (more so on a long train than a short one).
On other hills, the train is slowing down as the train begins to climb over the top, and speeding up again as it starts to descend again. Thus, if there is any airtime to be had before the top of the hill, it will be best in the front, as that is the part of the train moving fastest there, but airtime after the top of a hill will be best in the back. Thus, depending on the configuration of hills on a coaster, it may a better front seat ride or back seat ride.
Positive Gs from the bottom of a drop should be better in the middle of the train than either the front or back seat. This is because when the front of the train is at the bottom, it is still accelerating, while when the back is at the bottom, it is starting to slow down again. Thus, the train is moving fastest when the middle has reached the bottom. I have never noticed this myself, and haven't heard anybody else report it either. I suspect that there are a couple of reasons for this. First, the differences in speed are less pronounced than at the top of a hill (because there is less time for gravitational acceleration to affect the train when it is moving faster). Second, it is probably harder to tell the difference between, say, 3.4 and 3.6 Gs than it is to distinguish between -0.1 and 0.1 Gs.
Many people wonder whether coasters are safe. One often hears about people dying on rides, but more often than not, these are rumours rather than fact (tragically, people do sometimes die on rides, but overall, the safety record of coasters is admirable). From a physics standpoint, coasters are quite safe. For instance, in an inversion, the forces always conspire to keep the rider in the car. Coaster designers calculate the forces on the coaster to make it feel dangerous, but really be quite safe.
However, these calculations are done assuming the rider does nothing unusual. If you stand up in a sit-down coaster, the designer's calculations will no longer apply. Then, negative Gs may be enough to eject you. On a curve, your center of gravity may end up above the side of the car, and you will be in serious danger of being thrown out.
If treated with respect, coasters are quite safe. The chances are very slim that you will be hurt on a coaster if you ride it correctly. If you do something foolish, though, you greatly increase your chances of injury.
This work is copyright 1996 © by David A Sandborg
David holds himself responsible for all mistakes
contained within this work ... not that there are any ;-).
Please do not duplicate without prior permission from the author.
