Newton’s First Law of Motion

Question: A physics teacher provides the following statement of Newton’s first law of motion: Every body 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 on it. Explain two problems of this statement and rewrite it with the appropriate amendments.

This “examination question” is crafted based on an article by Williams (1999), titled Semantics in teaching introductory physics. He proposes two improvements in the above statement of Newton’s first law of motion: (1) the term “motion” should be replaced by “velocity” because motion does not have a quantitative definition; (2) the term “a net force” is more precise than “forces impressed” because a vector sum of forces acting on an object does not necessarily result in acceleration. However, there are still three problematic terms “isolated object,” “straight line,” and “inertial frame” which are commonly found in the modern formulation of Newton’s first law.

1. Isolated object: In Relativity: The Special and The General Theory, Einstein (1961/1916) states the law of inertia as “[a] body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line (p. 13).” That is, the law of inertia is applicable to an isolated object or free particle that is very far from other bodies. In other words, the term isolated object replaces the phrase “net force impressed on an object.” However, it is difficult to ensure an object to be completely free of external interactions such as gravitational forces and electromagnetic forces in an experiment. An isolated object is an idealized concept and it does not exist in the real physical world.

2. Straight line: In Motte’s translation of Newton's first law, it was stated that “every body perseveres in its state of rest, or of uniform motion in a right line (Newton, 1687/1995).” Currently, the term “straight line” is commonly used rather than “right line.” Importantly, a straight line may be defined as a path of a light ray (Poincaré, 1952). Thus, a straight line can be observed as a curve dependent on an observer’s frame of reference. Historically speaking, Galilei seems to be open to the idea of circular inertia. Furthermore, Newton (1687/1995) explains his first law by stating that “[t]he greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time (p. 19).”

3. Inertial frame: In The Science of Mechanics, Mach (1888/1989) writes that “I have remained to the present day the only one who insists upon referring the law of inertia to the earth and, in the case of motions of great spatial and temporal extent, to the fixed stars (p. 336-337).” On the other hand, Einstein (1961/1916) states that the laws of the mechanics of Galilei-Newton are valid only for a Galileian system of co-ordinates. However, current physics textbooks may specify this condition of validity as an inertial frame of reference. Thus, there is a problem of circularity if the inertial frame of reference is defined as the reference frame in which Newton’s first law holds. To resolve this problem, the inertial frame of reference can be defined as the reference frame in which space is homogeneous and isotropic, and time is homogeneous (Landau & Lifshitz, 1976).

How would Feynman answer?

Feynman’s father had an influence on him in understanding the concept of inertia. In the words of Feynman, “[m]y father taught me to notice things. One day, I was playing with an ‘express wagon,’ a little wagon with a railing around it. It had a ball in it, and when I pulled the wagon, I noticed something about the way the ball moved. I went to my father and said, ‘Say, Pop, I noticed something. When I pull the wagon, the ball rolls to the back of the wagon. And when I’m pulling it along and I suddenly stop, the ball rolls to the front of the wagon. Why is that?’ ‘That, nobody knows,’ he said. ‘The general principle is that things which are moving tend to keep on moving, and things which are standing still tend to stand still, unless you push them hard. This tendency is called ‘inertia,’ but nobody knows why it’s true.’ Now, that’s a deep understanding. He didn’t just give me the name (Feynman, 1988, p. 16).”

In short, Feynman prefers this general principle to be known as the “principle of inertia” rather than “Newton’s first law of motion” or “Galilei-Newton’s law of inertia.” It is possible that Feynman would explain the three problematic concepts: isolated object, straight line, and inertial frame as follows:

1. Isolated object: The principle of inertia can be stated as follows: “if an object is left alone and is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still.” That is, this law refers to an isolated object (or a free particle) that is completely undisturbed and has no potential energy at all. However, it is difficult to define a free particle or isolated object because there are gravitational fields everywhere. Importantly, by using the principle of least action, a free particle will move from one point to another at a constant speed such that the kinetic energy integral is least. If the particle were to go any other way, the velocities would be sometimes higher and sometimes lower than the average. The average velocity of the particle is the same for every case because it has to get from ‘here’ to ‘there’ in a given amount of time.

2. Straight line: One important aspect of the principle of inertia is the “physical” straight line of a moving particle. However, the concept of straight line can be ambiguous. It is possible to have different definitions of straight line on a plane, a sphere, or a hot plane. In general, the straight line can be defined as the shortest line between two points. Interestingly, the curve of shortest distance in space corresponds in space-time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the t-terms in relativity. Therefore, we should be aware of possible problems of defining a straight line. In essence, the concept of the straight line is dependent on an observer’s frame of reference.

3. Inertial frame: The idea that inertia represents the effects of interactions with faraway matter was first developed by Ernst Mach in the nineteenth century. Mach felt that the concept of an absolute acceleration relative to “space” was not meaningful; the usual absolute accelerations of classical physics should be rephrased as accelerations relative to the distant nebulae. In addition, an inertial frame of reference can also be automatically determined from the nebulae. However, a motion relative to the nebulae is a mysterious question that can be answered only by experiment. Thus, there are also difficulties in defining an inertial frame or reference frame which is moving with a constant velocity in a straight line.

In summary, Feynman would discuss the problems of defining isolated object, straight line, and inertial frame. However, he states that “if an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still (Feynman et al., 1963, section 9–1 Momentum and force).” Furthermore, Feynman (1995) explains that “[t]he principle of inertia is a statement that the time scale is independent of coordinates X; the classical trajectories are interpreted to follow the normal lines of constant phase (p. 72).” Essentially, Feynman has a deep understanding of the principle of inertia even though he says that he does not know the reason behind it (Feynman et al., 1963, section 7–3 Development of dynamics).

“Feynman’s answers” pertaining to isolated object, straight line, and inertial frame are based on the following compilation of his statements: 1. Isolated object: That is the principle of inertia—if something is moving, with nothing touching it and completely undisturbed, it will go on forever, coasting at a uniform speed in a straight line. (Why does it keep on coasting? We do not know, but that is the way it is.) (Feynman et al., 1963, section 7–3 Development of dynamics). Galileo made a great advance in the understanding of motion when he discovered the principle of inertia: if an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still (Feynman et al., 1963, section 9–1 Momentum and force). First, suppose we take the case of a free particle for which there is no potential energy at all. Then the rule says that in going from one point to another in a given amount of time, the kinetic energy integral is least, so it must go at a uniform speed. (We know that’s the right answer—to go at a uniform speed.) Why is that? Because if the particle were to go any other way, the velocities would be sometimes higher and sometimes lower than the average. The average velocity is the same for every case because it has to get from ‘here’ to ‘there’ in a given amount of time (Feynman et al., 1964, chapter 19 The Principle of Least Action).  But, as soon as we allow the presence of gravitating masses anywhere in the universe, concept of such truly unaccelerated motion becomes impossible, because there will be gravitational fields everywhere (Feynman, 1995, p. 93). 2. Straight line: Finally, our third bug — the one in Fig. 42-3 — will also draw “straight lines” that look like curves to us. For instance, the shortest distance between A and B in Fig. 42-6 would be on a curve like the one shown. Why? Because when his line curves out toward the warmer parts of his hot plate, the rulers get longer (from our omniscient point of view) and it takes fewer “yardsticks” laid end-to-end to get from A to B. So for him the line is straight — he has no way of knowing that there could be someone out in a strange three-dimensional world who would call a different line “straight” (Feynman et al., 1964, section 42–1 Curved spaces with two dimensions). The point of all this is that we can use the idea to define “a straight line” in space-time. The analog of a straight line in space is for space-time a motion at uniform velocity in a constant direction. The curve of shortest distance in space corresponds in space-time not to the path of shortest time, but to the one of longest time, because of the funny things that happen to signs of the t-terms in relativity. “Straight-line” motion—the analog of “uniform velocity along a straight line”—is then that motion which takes a watch from one place at one time to another place at another time in the way that gives the longest time reading for the watch. This will be our definition for the analog of a straight line in space-time (Feynman et al., 1964, section 42-4 Geometry in space-time). Clearly, the vector (dxμ/ds) along the geodesic represents a tangential velocity, Δtμ, along the geodesic, which is the “physical” straight line (Feynman, 1995, p. 130). 3. Inertial frame: The idea that inertia represents the effects of interactions with faraway matter was first developed by Ernst Mach in the nineteenth century, and it was one of the powerful ideas that Einstein had in mind as he constructed his theory of gravitation. Mach felt that the concept of an absolute acceleration relative to “space” was not meaningful; that instead, the usual absolute accelerations of classical physics should be rephrased as accelerations relative to the distant nebulae (Feynman, 1995, p. 70).  We shall now show that the inertial frame is now also automatically determined from the nebulae, and the phenomena of inertia for accelerations relative to the nebulae can be understood if the “length determining principle” is accepted (Feynman, 1995, p. 72). In special relativity, extensive use is made of reference frames which are moving with a uniform velocity in a straight line (Feynman, 1995, p. 92).  Then we say to him, “Now, my friend, is it or is it not obvious that uniform velocity in a straight line, relative to the nebulae should produce no effects inside a car?” Now that the motion is no longer absolute, but is a motion relative to the nebulae, it becomes a mysterious question, and a question that can be answered only by experiment (Feynman et al., 1963, section 16–1 Relativity and the philosophers).

Note: 

1. An example of examination question pertaining to Newton’s first law of motion is shown below: A student makes the following statements of Newton’s laws of motion: “First Law: Every body continues in its state of motion unless it is acted upon by a resultant external force.”… The statement of the First Law is incomplete in two aspects. Identify the two aspects in which it is incomplete and hence rewrite it with the appropriate amendments (Council for the Curriculum, Examinations & Assessment, 1998). 

2. You may want to take a look at this website: 

http://physicsassessment.blogspot.sg/2016/05/ib-physics-2015-higher-level-paper-2_25.html

References:

1. Einstein, A. (1961/1916). Relativity: The Special and The General Theory. New York: Random House.
2. Feynman, R. P. (1988). What Do You Care What Other People Think? New York: W W Norton.
3. Feynman, R. P. (1995). Lectures on gravitation (B. Hatfield, Ed.). Reading, MA: Addison-Wesley.
4. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
5. Feynman, R. P., Leighton, R. B., & Sands, M. L. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.
6. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.
7. Mach, E. (1888/1989). The Science of Mechanics - A Critical and Historical Account of its Development. La Salle: Open Court.
8. Newton, I. (1687/1995). The Principia. Translated by Andrew Motte. New York: Prometheus.
9. Poincaré, H. (1952). Science and hypothesis. Mineola, NY: Dover.
10. Williams, H. T. (1999). Semantics in teaching introductory physics. American Journal of Physics, 67(8), 670–680.