Trajectories

Type of physical science: Classical physics

Field of study: Mechanics

The trajectory of a particle can be predicted using Newton's second law of motion if the mass of the particle and the total force acting on it are known.

Overview

The trajectory (path) of a moving object is dictated by Sir Isaac Newton's second law of motion, first published in PHILOSOPHIAE NATURALIS PRINCIPIA MATHEMATICA (1687; MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, 1729), which states that a body of mass m will undergo an acceleration a when an external force F acts on it. (An external force is one that does not act between two components of the mass itself.) Newton's second law is concisely stated as F = ma (where F is measured in newtons, m in kilograms, and a in meters/second²); however, it does not directly indicate either the shape of the trajectory or where the object will be at some future time. These are the subjects of trajectory analysis, which generally requires the use of the calculus and a detailed knowledge of the net force acting on the mass.

Newton's second law applies only to a given particle of mass (a geometric point without measurable dimensions). It is always possible to model a real object as a particle of equivalent mass placed at the object's center of mass (the average location of all the body's mass). In addition, the quantities F and a are vectors: Each has both a magnitude and a direction. When several forces are present, they are added together as vectors to give the single net force that acts on the center of mass of the object. Each force vector generally has components along each of three mutually perpendicular directions, and all the components along a particular direction are added to give the component of the net force along that direction. If a particle experiences a force of constant magnitude and direction, then it accelerates in the same direction as the force and with a constant acceleration. If the force is changing (either in direction or magnitude), then at each instant in time, the particle's acceleration will be in the direction of the force at that instant and will have a magnitude determined by Newton's second law.

The quantity acceleration is the rate at which the object's velocity (speed and direction) is changing. In order to compute a particle's trajectory, it is necessary to know its initial velocity.

A positive acceleration along a given direction means that the object is moving with ever-increasing speed (in that same direction). A negative acceleration indicates that the object is slowing down, or even starting to move in the opposite direction. A further consequence of Newton's second law is that an object which is already moving at some constant speed cannot change its velocity (that is, experience any type of acceleration) unless an external force acts on it. Thus, the simplest of all types of trajectories is a straight line. For example, a mass which has started to move in some direction in deep space, far from the significant gravitational attraction of stars or planets, will not experience any forces. Therefore, the mass will continue to move in a straight line along the original direction until it does encounter some force, such as the result of a collision with another mass. On the earth, the object would typically experience not only the force of gravity but also forces that are attributable to friction, such as atmospheric drag forces or contact friction while sliding along a surface. These forces will tend to decrease the object's speed by removing energy from it, and they may change the shape of its trajectory as well.

Because the relation F = ma indicates the rate at which a body's velocity is changing over time, it is called a differential equation. In general, such an equation requires special methods of calculus for its solution and indicates where an object will be at any future time. Indeed, another useful way to view a trajectory is as the set of all positions that an object has during its motion.

An interesting property of trajectories is that their shape depends on the frame of reference in which they are viewed. For example, if a mass is dropped inside a train car that is moving with constant speed, then the passengers will see the mass fall straight downward. This motion results from the fact that both the passengers and the mass all have the same horizontal speed; gravity is the only external force present, and it causes the purely vertical motion. To an observer standing on the ground and looking through a window into the train, however, the object moves forward and downward on a curved path. This motion is seen because, relative to the ground, the object has some initial horizontal speed. The difference in trajectory shape in this example is relatively simple. A more complex difference occurs when one observer is in a rotating frame of reference. For example, a cannonball fired horizontally at high speed from the North Pole actually moves along a path which would otherwise be straight except for the force of gravity causing it to follow the curvature of the earth. As viewed from a stationary point in space, the trajectory is due south. As the cannonball flies, however, the earth rotates underneath it, giving the appearance to an observer on the ground that the cannonball is curving to the west.

Because of the relatively slow rotation rate, the earthbound observer does not recognize intuitively what is causing the deflection of the trajectory to the west and concludes that there must exist a special force not associated with gravity which is causing this effect (known as Coriolis acceleration).

Applications

Trajectories are best illustrated by several examples that show different behaviors of motion. An object released from a high platform will fall straight downward (neglecting any horizontal wind) with increasing speed. The simplest model of this motion assumes that there is no atmospheric friction (drag), in which case the speed of the body increases constantly according to the equation v = gt, where v is the velocity in meters/second, g is the acceleration caused by gravity (9.81 meters/second²), and t is the elapsed time in seconds. The distance d through which the body falls is found from the equation d = (1/2)gt². A more sophisticated model would include the effects of atmospheric drag, which would not change the purely vertical shape of the trajectory but could significantly alter the speed and distance formulas, depending on the particular object's shape.

A more interesting trajectory results from casting the mass from a high platform, with its velocity initially in the horizontal direction. Gravity immediately begins to pull the object downward but, according to Newton's second law, the velocity change is only in the direction of the external force (downward). Therefore, the horizontal speed of the body does not change, again neglecting air friction. As a result, the trajectory is the familiar downward curve that is characterized mathematically as a parabola. This same type of trajectory is also produced by a projectile fired from the earth's surface with both horizontal and vertical components to its initial speed. The body arcs upward, with gravity acting only to slow its vertical motion, while its horizontal speed remains constant. Eventually, the projectile reaches the apex of its trajectory and then starts to fall back. If air friction is significant, then the shape of the trajectory is not precisely parabolic, but still qualitatively similar.

It is also possible to apply a force to a particle in such a way that only the direction of motion changes while the speed remains constant. This situation is easily demonstrated by swinging a small object at the end of a string in a horizontal circle about one's head. By keeping a constant tension in the string (that is, by applying a constant magnitude of force), the object can be kept moving in a circle at constant speed. Its direction of motion is constantly changing, however, as predicted by Newton's second law. The direction of the force is always from the object back toward the center of the circle. An important example of such a trajectory is that of subatomic particles being forced to move in a circular path by means of a magnetic field, which is a method used by nuclear physicists in particle-collision experiments. Because of their motion in the magnetic field of the accelerator, these electrically charged particles experience what is known as the Lorentz force, which is always directed toward the center of a circular path. The magnetic field holds the particles on the path, their energy is increased, and they are allowed to collide with other particles as part of the experiment being performed.

Satellite trajectories (orbits) are even more complex. The simplest is a circular path, such as about the earth, on which the satellite moves at constant speed, with gravity providing exactly the right force to change only the direction. If the orbit is high enough, generally greater than 1,000 kilometers, then no atmospheric drag is present and the satellite will continue on the circular trajectory forever. To initiate such a trajectory, the booster rocket for the satellite must stop its engines just as it reaches the condition v² = GM/r, where v is the speed (in meters/second), r is the distance (in meters) from the center of the earth, and GM is the product of the universal gravitational constant (6.67 x 10^-11 meters³/kilograms - seconds²) and the earth's mass (5.98 x 10²⁴ kilograms). If the satellite is moving somewhat faster or slower than the particular speed needed for the given circular radius, then the resulting trajectory is an ellipse with the earth's center located at one focus. The gravitational force from the earth still regulates the motion, but in this case, the satellite will move alternately closer to and farther from the earth, with its speed becoming respectively greater and then lesser. Special terminology used for this most common type of satellite orbit includes "apogee," which is the point on the orbit farthest from the earth, and "perigee," which is the point nearest to the earth. If the perigee is located at least 1,000 kilometers above the earth's surface, then no significant atmospheric drag occurs and the satellite will never alter from its orbit.

Two other types of satellite trajectories are possible, both of which result in the satellite leaving the earth and never returning. If the initial speed of the satellite at injection is such that v² = 2GM/r, then the resulting trajectory will be a parabola with a perigee close to the earth but with its two open ends an infinite distance away. This situation is the simplest of all escape trajectories and must be used to start a space probe on a mission to another planet. If the initial speed is greater than that needed to produce the parabola, then the resulting path is a hyperbola. The only qualitative difference between parabolas and hyperbolas is that the two legs of the parabola eventually become parallel at great distances from the earth while those of a hyperbola do not. As for the spacecraft's motion, it will be greater (on the average) on a hyperbola than on a parabola; that is, the spacecraft has more than enough energy to get away from the earth without gravity slowing and eventually pulling it back.

For satellites in relatively low orbits (circular or elliptical), the effects of atmospheric drag on the trajectories can be quite pronounced. Qualitatively, the drag reduces the satellite's energy, causing it to fall steadily closer to the earth in a form of spiral motion. Unless corrective action is taken, such as using the satellite's own engines to counteract the drag and raise it back to a higher altitude, the spacecraft will eventually burn up from the intense heat of friction and crash on the earth.

Context

The motion of objects has been a subject of fascination since antiquity, beginning with the first awareness of the apparent motion of the stars. Much of the early effort in studying motion attempted to characterize how objects moved, without considering the causes. For example, Plato believed that each celestial body must follow a circular path, since that was the perfect trajectory. Aristotle modeled the universe as a set of natural places, with the earth being in the lowest natural place. He suggested that every object moved toward its natural place.

Unfortunately, this view was not subject to verification in the sense that it did not predict in detail how a body moves, that is, what its position would be at some future time. It was not until the Middle Ages that serious attempts to quantify motion were made. Galileo Galilei (1564-1642) was the first to understand that the trajectory of a body is determined by both its speed and its direction of motion. His experiments and writings about motion brought considerable precision to the human understanding of how objects move under the influence of gravity.

Newton's formulation of the laws of motion for particles remains the greatest single advance in knowledge of the mechanical world. These laws provide a means to predict or understand the trajectory of a particle in the presence of any type or combination of forces.

Further, Newton's invention of differential and integral calculus (which was also done simultaneously and independently by Gottfried Wilhelm Leibniz) was and still remains indispensable for predicting and analyzing trajectories.

In everyday applications, where the masses and forces are within the realm of human experience, Newton's laws are adequate for such analysis. Near the end of the nineteenth century, however, it was found that, if exceedingly small (atomic-sized) or large (stellar-sized) masses are involved or if the speeds between frames of reference are significant fractions of the speed of light, then more sophisticated theories are required to describe the trajectories with any accuracy.

These situations led to the development of quantum mechanics and the special and general theories of relativity, which came to be used routinely by scientists for the study of those environments.

Principal terms

ACCELERATION: the rate at which an object's velocity changes in speed and/or direction

APOGEE: the point of an orbit at which the object is the farthest from the earth

FORCE: the amount of push or pull exerted against an object

MASS: the amount of matter in an object

PERIGEE: the point of an orbit at which the object is the nearest to the earth

VELOCITY: the speed and direction of a moving object

Bibliography

Asimov, Isaac. UNDERSTANDING PHYSICS. Vol. 1. New York: Walker, 1966. This two-volume set treats both classical and modern physics using almost no mathematics. The explanations are lucid and witty, which is one of Asimov's hallmarks. His examples are drawn from common experience and serve to illustrate the concepts quite effectively.

Barbour, Julian B. ABSOLUTE OR RELATIVE MOTION? Vol. 1 in THE DISCOVERY OF DYNAMICS. Cambridge, England: Cambridge University Press, 1989. Primarily a philosophical study of the basic problem of frames of reference, but also contains many interesting historical insights.

Brancazio, Peter J. "Playing It by Ear." SCIENTIFIC AMERICAN 248 (April, 1983): 76. Although brief, this article contains a number of useful findings about the effects of atmospheric drag on the trajectory of a baseball, as well as how an outfielder uses different sensory information to predict intuitively where the ball will land.

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. Vol. 1. Reading, Mass.: Addison-Wesley, 1963. These lectures were intended to challenge some of the brightest physics students at the California Institute of Technology, but they have exceptional clarity and appeal for general audiences. One of the few books that effectively treat the approximate nature of physical laws.

Frautschi, S. C., R. P. Olenick, T. M. Apostol, and D. L. Goodstein. THE MECHANICAL UNIVERSE: MECHANICS AND HEAT. New York: Cambridge University Press, 1986. The companion text to THE MECHANICAL UNIVERSE film series (Public Broadcasting System), which is based on Goodstein's lectures at the California Institute of Technology. The historical development of these subjects, carefully integrated with the analysis, is one of the best treatments available.

Gamow, George. MATTER, EARTH, AND SKY. Englewood Cliffs, N.J.: Prentice-Hall, 1958. Written as a textbook for introductory physics courses, this work is based on Gamow's long experience in lecturing and writing for general audiences interested in science. Successfully combines physics, chemistry, geology, and astronomy, including numerous helpful diagrams and some of the author's own cartoons.

Newton, Isaac. MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY. Translated by Andrew Motte and edited by Florian Cajori. Berkeley: University of California Press, 1934. In this book, Newton puts forth his laws of motion and the law of universal gravitation. While the language is flowery, the mathematics are fairly easy to follow: Newton presents his theories using geometry, making the developments sometimes tedious but always comprehensible.

Rothman, Milton A. THE LAWS OF PHYSICS. New York: Basic Books, 1963. Using a minimum of mathematics, this book introduces the basic principles of classical and modern physics, with many excellent examples. Gives considerable attention to the philosophical aspects of the human knowledge of physical reality.

Centrifugal/Centripetal and Coriolis Accelerations

Numerical Solutions of Differential Equations