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Coaster Dynamics
Physics Primer
Chapter 7
Newton's First Law of Motion
Sir Isaac Newton
If the field of science were like the worlds of sports and music, Isaac Newton would be a true superstar.
In addition to his seminal work in classical mechanics, he also originated the Law of Universal Gravitation (which overthrew the previous ideas of astronomy), independently developed the mathematics of calculus (derivatives and integrals) to facilitate the analysis of his laws of motion, and was an accomplished mathematician and scholar.
Prior to Newton, the most significant ideas about the science of motion came from Aristotle and Galileo. The philosopher Aristotle, and others that followed him, originally suggested that the "natural state" of objects was at rest. They theorized that to make an object move, an external agent, which they called a "force" was needed. This view was dominant for thousands of years... until Galileo changed it. Through careful experimentation, Galileo showed that the motion of an object would remain constant in the absence of a force, rather then go to zero. In other words, he theorized that an object will naturally resist changes in motion, and Galileo called this property "inertia." However, Galileo did not offer a way to quantify this new property he proposed.
Isaac Newton was born in 1642, the same year Galileo died. Following in his footsteps, Newton refined and extended Galileo's idea of inertia -- leading to his famous Laws of Motion. Newton probably developed his laws of motion during his early twenty's, but did not formally present them until the age of 44, when he published the book, Principia Mathematica Philosophiae Naturalis, in 1686.
Newton's First Law of Motion
In his First Law of Motion, Newton expanded Galileo's idea of inertia, adding that in the absence of a force, the motion of an object will remain unchanged, and specified how the motion would continue.
In the time of Isaac Newton, the language of mathematical equations we use now was not yet invented. Thus, Newton's Principia was published using the written word of the day... in Latin. Here is the translation of Newton's words:
Newton's First Law of Motion:
Every material object persists in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.More than just a restatement of inertia, the First Law of Motion, states that with no forces acting, uniform motion will continue in a straight line. The concept of "natural" motion in either a straight line, or in a circle, was a puzzle to the scientists at the time. In particular, scientists and astronomers pondered about the "natural" motion of objects observed on earth (which was often in straight lines) -- and of the planets (which it was recently discovered, traveled in circular orbits). The correct explanation for these "natural" motions wasn't understood until Newton developed his laws of motion.
Although the concept didn't explicitly exist at the time, Newton's First Law of Motion can also be considered a result of, or a restatement of, the Equivalence of Reference Frames -- a basic tenet of today's physics. This principle states that the laws of physics must remain valid regardless of the reference frame of measurement.
If the "natural" motion of an object were at rest, this would violate the equivalency of reference frames. For instance, if we found an object to be at rest in one reference frame, and then measured the motion from another reference frame that was moving at a constant velocity with respect to the first reference frame... then the second measurement would indicate a non-zero velocity -- in contradiction to the first measurement and our assumed (incorrect) law of motion.
Furthermore, consider the case of measuring the velocity in one reference frame, and then standing on our head and measuring the velocity in that reference frame. In the second (upside-down) reference frame, we would find the object moving in the opposite direction. In order to reconcile the difference, the correct law of motion must say that the motion will continue in a straight line. That is, as long as we maintain a fixed reference frame, the motion will remain unchanged in a straight line -- as we see it. Someone in a different reference frame will see the motion also continue in a straight line -- although in a different direction compared to our observations.
Also implicit from Newton's First Law of Motion is the fact that the direction components of motion are independent.
Using reasoning similar to the "standing on our head" reference frame example above, it can be deduced that the x-component of motion of an object obeys the laws of motion, just as does the y-component of motion (or the z-component... and another other component you wish to use). Thus, no matter what the reference frame, the motion is equivalent... and it can be considered the sum of the individual x, y, and z components of motion for the reference frame of choice.
Projectile Motion
Despite the seeming simplicity of Newton's First Law of Motion, its implications are quite powerful, and it can be used to analyze a number of interesting problems. A particularly interesting analysis for roller coaster designers is that of projectile motion.
Consider the case of driving a car along a horizontal road and then suddenly driving off a cliff -- plunging down into a canyon. When the car flies over the edge, its motion becomes that of a projectile, and the path the car takes through the air is called a projectile trajectory. As you might imagine, the thrill of driving off a cliff (without the danger of actually crashing, of course) is something roller coaster designers would like to simulate... so the analysis of projectile motion is of great interest to them.
The case being analyzed is illustrated in Figure 7-A.
Figure 7-A. Projectile Motion.
Our objective is to determine what shape the projectile trajectory will be.
We begin by defining a coordinate system with the positive x direction parallel to the path of the initial car motion, and the positive y direction upward. The axes origin is at the center of gravity of the car, at the point the car leaves the cliff. With this coordinate system, the initial velocity is vx,0, and vy,0 = vz,0 = 0. At the moment the car goes over the edge, the position is at x0 = y0 = z0 = 0.
From Newton's First Law of Motion, as discussed above, we know that the velocities in the coordinate directions are independent, and constant if there are no forces acting in that direction. The only force on the car is gravity, acting in the -y direction. Thus, we conclude immediately that vz is always zero -- so the motion is limited to the x-y plane.
Similarly, we conclude that in the x direction, the velocity is a constant, equal to vx. Thus, the x position is equal to:
(7-1) x = (vx)(t).
From the analysis of Free Fall Motion in Chapter 5, we know that the y velocity will be equal to:
(7-2) vy = vy0 + (ay)(t) = -(g)(t)
and the y position is equal to:
(7-3) y = y0 + (vy0)(t) + (1/2)(ay)(t²) = -(1/2)(g)(t²).
Combining equations (7-1) and (7-3), and eliminating t, yields:
(7-4) y = -(1/2)(g)(x/vx)² = -(g/(2vx²))(x)².
Finally, simply rearranging equation (7-4) yields:
(7-5) x² = (-2vx²/g)(y),
where -2vx²/g is a constant.
Any math wizards reading this (surely there must be some... although I know I'm not one) will recall that the general equation for a parabola is given by:
(7-6) x² = (4c)(y).
In the case where c is a negative constant, this will be a parabola with its vertex at the origin, and its focus at the point (0, -c).
So... by comparing equations (7-5) and (7-6), we conclude that the projectile trajectory is a parabola.
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