Coaster Dynamics: Physics Primer Chapter 1



Coaster Dynamics
Physics Primer

Chapter 1

Introduction:   From Falling Apples to Roller Coasters

According to a legendary story, a young man in his early 20's named Isaac Newton was deep in thought one day when he saw an apple fall from a tree. Whether or not this was really the genesis of a revolution in the science of physics, it is a fact that in 1686, 44 year old Isaac Newton published the book, Principia Mathematica Philosophiae Naturalis, which completely overturned the previous theories in physics and formed the foundation of the science for the next 200 years. Along with Albert Einstein, whose ideas in his Special Theory of Relativity started another revolution in physics in 1905, Sir Isaac Newton stands among the most influential scientists in history.

The branch of physics known as Newtonian Mechanics provides the framework for the study of particle motion. Newton's Laws of Motion are the basis for the analysis and understanding of roller coaster motion -- and despite their deceptively simple nature, are the underpinnings of a vast array of theories and applications which powered scientific development pre-dating the start of the Industrial Revolution to the advent of space travel.

Before delving into details, it is useful to take a step back and consider the question:   What is the science of physics?

Physics is a branch of science devoted to the study and analysis of the physical world. In the early days, this meant the study of the world we can easily see and measure in our everyday lives. This branch of physics is called Classical Physics. Later, so-called Modern Physics expanded to include situations and phenomenon not easily observable by our normal senses -- for instance, models of the universe, phenomenon at relativistic (near the speed of light) speeds, and the sub-atomic world. Much of the development of Modern Physics evolved from ideas proposed by Albert Einstein, and by the so-called Quantum Theory developed by others later. Newtonian Mechanics is the subset of Classical Physics devoted to the study of the motion of objects at non-relativistic speeds.

The objective of physics is to understand, and to develop mathematical models of the world and the universe. Implicit in this activity is the ability to predict the behavior of the world, based on the mathematical models.

Newtonian Mechanics and State-Determined Systems

The analysis of the motion of objects, encapsulated by Newton’s Laws of Motion, forms a mathematical model of a "system." A system is a loosely defined thing which, in this case, includes an object and any associated objects or phenomenon which exert forces on the object. It is assumed that the system can be clearly identified, and items in the system can be distinguished from items not in the system. Based on the laws of motion, one can then predict how the system will behave.

This type of system, which is basic to all scientific study, is called a "state-determined" system. A key characteristic of a state-determined system is that its behavior can be accurately predicted based solely on its current "state." The future of the system can be predicted solely from the present, without regard to the past. Also implicit from this characteristic is that the past cannot be ascertained from the present.

For example, at any moment in time, a roller coaster car will be traveling at a particular speed and direction. Based on that information, along with any known forces acting on it (from the roller coaster track or gravity), its future motion can be calculated. It does not matter how the car reached its current speed -- only what the current speed and forces are. Furthermore, knowing only the current speed and forces, you cannot determine what previously happened to the car that resulted in the current state.

Another key characteristic of a state-determined system is that its behavior depends only on certain key physical quantities, called "state variables." All other related physical quantities describing the system can be derived from the state variables.

The choice of state variables is somewhat arbitrary. For example, in the case of the motion of an object, one might choose force and mass as the state variables (as embodied in Newton's famous Second Law of Motion). However, the more fundamentally preferred state variables are momentum (the product of mass times velocity) and the net force acting on the object. What is not arbitrary, however, is that for any given system, there is a minimum number of state variables required to precisely model it. For the motion of a roller coaster car, there are 2 state variables required (and consequently, it is called a "second-order" dynamic system).

Why is it important that the model of a roller coaster is a valid state-determined system? Because it means the laws of motions can be independently applied to the analysis of individual elements of a track. For instance, if designing a loop that is part of a complex track, all that must be known is the speed of the car when it enters the loop. It doesn’t matter what happened to the car before arriving there. In the days before the invention of computers, roller coaster tracks were designed by analyzing individual track elements using simple handwritten equations. This made possible the design of safe roller coasters despite the relatively simple methods available at the time.

At first thought, one might think that all systems are state-determined -- but many are not. For instance, the stock market is a "system" whose behavior is influenced by many factors. However, no one fully understands what causes stock prices to move up or down. So, as far as anyone knows, the stock market is not a state-determined system -- and its behavior cannot be mathematically modeled and accurately predicted. Perhaps someday a future genius will develop a valid model of the stock market, but thus far, many people have tried -- and lost fortunes -- trying to do so. Good thing roller coasters don’t behave like the stock market... or your grandparents might not have survived that thrilling ride on Loop-the-Loop at Coney Island in 1901!

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