Section 9–1 Momentum and force

(Newton’s First Law / Momentum / Force)

In this section, the three interesting points discussed are Newton’s First law of dynamics, momentum, and force.

1. Newton’s First Law:

“The First Law was a mere restatement of the Galilean principle of inertia just described (Feynman et al., 1963, section 9–1 Momentum and force).”

Feynman states the principle of inertia as “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.” He explains that this law never appears in nature because a sliding block will eventually stop. In essence, Newton’s First Law of dynamics is developed by Galileo’s imagination. Simply put, Newton’s First Law is based on idealizations and it cannot be directly (or exactly) observed in nature. Similarly, Eddington (1928) rephrases Newton’s First Law as “[e]very body continues in its state of rest or uniform motion in a straight line, except in so far as it doesn’t (p. 124).” Curiously, Feynman earlier (Volume I, Chapter 7) says that we do not know why an object coasting at a uniform speed in a straight line.

One may not agree with Feynman that the First Law was a mere restatement of the Galilean principle of inertia. Strictly speaking, Galileo did not explicitly state a general principle of linear inertia. On the contrary, Galileo suggests a concept of circular inertia: “a ship … would move continually around our globe without ever stopping and placed at rest it would perpetually remain at rest, if in the first case all extrinsic impediments could be removed, and in the second case no external cause of motion were added (Galilei, 1613, pp. 113–114.)” In other words, an object may continue in its state of circular motion unless there is an (external) resultant force. Perhaps Galileo would prefer this modern version of the law of inertia: “A free object continues in its state of rest or moves along a geodesic in spacetime.”

2. Momentum:

“Now the momentum of an object is a product of two parts: its mass and its velocity (Feynman et al., 1963, section 9–1 Momentum and force).”

Feynman mentions that a lot of words in physics have precise meanings in physics. He defines the momentum of an object as a product of its mass and its velocity. However, this is not a general definition of momentum. In the special theory of relativity, the momentum of a fast moving particle (p = γmv) includes a Lorentz factor, γ. In quantum physics, the momentum of a photon (p = h/λ) is equal to Planck’s constant divided by its wavelength. Alternatively, the momentum of electromagnetic radiations (p = E/c) can be calculated by the total energy of electromagnetic radiations divided by the speed of light. To be more precise, we should adopt the term linear momentum that is distinguished from angular momentum.

According to Feynman, the Second Law gives a specific way of determining how the velocity changes under different forces and the Third Law is essentially action equals reaction. However, Newton’s three laws of dynamics (or motion) can be consistently related to the linear momentum. We can rephrase the First Law as “a free particle always moves with a constant linear momentum relative to an inertial frame of reference. The Second Law can be more precisely stated as “the rate of change of linear momentum of a particle with respect to time is proportional to the force acting on it”. The Third Law can be related to the principle of conservation of linear momentum: the linear momentum of a system is constant if there is no external resultant force acting on the system.

3. Force:

“As a rough approximation, we think of force as a kind of push or pull that we make with our muscles, but we can define it more accurately now that we have this law of motion (Feynman et al., 1963, section 9–1 Momentum and force).”

Feynman elaborates that Newton’s Second Law may be written mathematically as F=d(mv)/dt and if the mass of an object is constant, it can be simplified as F = ma. This relationship does not only stipulate changes in the magnitude of the momentum and velocity but also in the direction. That is, the direction of the change in the momentum and velocity is the same as the direction of the force. Students should realize that acceleration, or a change in a velocity, has a wider meaning than its use in daily language: when an object slows down, we say it accelerates with a negative acceleration. However, Feynman in chapter 12 adds that if we insist upon a precise definition of force, we will never get it! This is because the Second Law is not exact and it involves approximations and idealizations.

Note that Newton did not specifically write the equation F = ma. In fact, Newton’s second law may be known as Euler’s First Law because Euler (1736) first develops the “F = ma” scheme and extends it to the motion of rigid bodies. Interestingly, Wilczek (2004) expresses his difficulties in learning F = ma and writes that “Newton’s second law of motion, F = ma, is the soul of classical mechanics. Like other souls, it is insubstantial. The right−hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left−hand side, on the other hand, has no independent meaning. Yet clearly Newton’s second law is full of meaning… (p. 11).”

Questions for discussion:

1. Is Newton’s First Law of dynamics a mere restatement of the Galileo’s principle of inertia?

2. Is there a general definition of linear momentum? (The linear momentum of an object is the ability to generate an impulse over a period of time?)

3. What are the meanings of Newton’s Second Law of dynamics as expressed by F = ma?

The moral of the lesson: Newton’s First Law of dynamics is related to Galileo’s method of idealization and this law cannot be strictly observed in nature.

References:

1. Eddington, A. (1928). The Nature of the Physical World. New York: Cambridge University Press.

2. Euler, L. (1736). Mechanica sive motus scientia analytice exposita. Saint Petersburg: Press of the Academy of Sciences.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Galilei, G. (1913). Letters on Sunspots (translated by S. Drake). In G. Galilei (1957). Discoveries and Opinions of Galileo. New York: Doubleday.

5. Wilczek, F. (2004). Whence the force of F= ma? I: culture shock. Physics Today, 57(10), 11-12.

Section 8–5 Acceleration

(Defining acceleration / Determining acceleration / Parabolic motion)

In this section, the three interesting points discussed are a definition of acceleration, determination of acceleration, and parabolic motion under constant acceleration.

1. Defining acceleration:

“Acceleration is defined as the time rate of change of velocity (Feynman et al., 1963, section 8–5 Acceleration).”

According to Feynman, the next step in developing the equations of motion of an object is to introduce another idea by asking the question, “How does the velocity change?” He cites an interesting example that is related to the great excitement about some cars can move from rest to 60 miles an hour in ten seconds. By using this example, we can understand how fast the velocity changes per second or the concept of average acceleration of an object. However, Feynman does not discuss common equations of motion that are based on the concept of constant acceleration. The equations of motion are sometimes expressed as follows: s = ut + ½ at², v = u + at, and v² = u² + 2as.

Feynman defines acceleration as the time rate of change of velocity and adds that we can write the acceleration in terms of the derivative dv/dt. In other words, acceleration is defined as the rate of change of velocity of an object with respect to time. To be precise, we can use the expressions Dv/Dt and dv/dt to represent average acceleration and instantaneous acceleration respectively. On the other hand, Feynman explains that accelerations are usually not constant, but it is constant in the example provided because the force on the falling body is constant. (Newton’s second law stipulates that the acceleration is proportional to the force, but this will be covered in the next chapter.) In short, the acceleration is constant because of simplifications and idealizations of the gravitational force near the surface of the Earth.

2. Determining acceleration:

“Since velocity is ds/dt and acceleration is the time derivative of the velocity, we can also write a = d/dt(ds/dt) = d²s/dt² (Feynman et al., 1963, section 8–5 Acceleration).”

One may expect Feynman to use an experiment to determine the acceleration of an object. Instead of using the experiment, he simply determines the acceleration by applying the rules of calculus or differentiation. That is, the acceleration is the time derivative of the velocity, and thus, we can write a = dv/dt = d²s/dt². Next, it may be surprising that Feynman mentions that we have a “law” in which the velocity is equal to the integral of the acceleration. However, one may prefer using the phrase “mathematical relationship” over “law” and elaborate that the distance can be determined by integrating the acceleration twice with respect to time.

More important, the acceleration of an object can be determined experimentally by first measuring its velocities. For example, we can use an odometer or a global position system speedometer. As a result, we can determine the acceleration by calculating the slopes of many points in a graph of velocity with respect to time. Alternatively, one may use a high-speed video to record the motion of an object and use Tracker Video Analysis App to determine the acceleration of the object. If we are the moving object, we can measure our speed by using a smartphone that has a global positioning system receiver. The use of a built-in accelerometer in the smartphone may not be accurate because it may measure “net g-force” instead of acceleration (Vogt & Kuhn, 2012).

3. Parabolic motion:

“When this equation is plotted we obtain a curve that is called a parabola; any freely falling body that is shot out in any direction will travel in a parabola (Feynman et al., 1963, section 8–5 Acceleration).”

Feynman suggests that a three-dimensional motion can be first illustrated on a two-dimensional diagram in terms of an x-distance and a y-distance before it is extended to three dimensions. The extension of the motion to three dimensions requires an axis that is perpendicular to the first two axes, and it can be labeled as the z-distance. The velocity in the first two dimensions during an interval can be approximated by letting Δt go to 0 and expressed as: v = ds/dt = √(dx/dt)²+(dy/dt)² = √(v_x² + v_y²). One may clarify that this equation is based on the assumption of Euclidean geometry. Currently, physicists opine that the true geometry of spacetime is non-Euclidean geometry as required by Einstein’s general theory of relativity.

In projectile (parabolic) motion problems, students can first assume an object moves horizontally with a constant velocity u, and at the same time moves vertically downward with a constant acceleration –g. The relationship established between y and x can be considered as the equation of the motion of the moving object. To understand better, one may include Feynman’s explanation in a later chapter as follows: “in other words, motions in the x-, y-, and z-directions are independent if the forces are not connected (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).” In essence, the three-dimensional motions of the object can be resolved into perpendicular directions that are independent of each other. Thus, the equations in terms of x, y, and z, are sometimes known as independent equations.

Questions for discussion:

1. Should an acceleration of an object be defined time rate of change of velocity?

2. Should acceleration be determined mathematically or measured experimentally?

3. Why are the equations of motion of an object in x-direction and y-direction independent of each other?

The moral of the lesson: the three-dimensional motion of an object can be first expressed in terms of an x-distance and y-distance, and the motions of the object in the x-, y-, and z-directions are independent of each other.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Vogt, P., & Kuhn, J. (2012). Analyzing free fall with a smartphone acceleration sensor. The Physics Teacher, 50(3), 182-183.

Section 8–4 Distance as an integral

(Distance in terms of infinitesimals / Integration process / Definite functions)

In this section, the three interesting points discussed are the distance in terms of infinitesimals, integration process, and definite functions.

1. Distance in terms of infinitesimals:

“We use the same idea, and express the distance in terms of infinitesimals. Let us say, ‘In the first second her speed was such and such and from the formula Δs = vΔt…’ (Feynman et al., 1963, section 8.4 Distance as an integral).”

Feynman discusses the inverse problem: how can we deduce the distance if the speed is known? He suggests the idea of expressing the distance in terms of infinitesimals. In a sense, there are some linguistic issues with regard to the words distance and infinitesimals. To be precise, physics teachers should distinguish “distance traveled” and “distance from the origin” (or position). Furthermore, the formula Δs = vΔt as cited in the next sentence is not definitely in terms of infinitesimals. Strictly speaking, this is possible if there is a limiting condition in which Δt approaches zero. Instead of the formula Δs = vΔt, Feynman could state ds = v dt and explain that the infinitesimal ds is the smallest change in displacement (v means the instantaneous velocity).

Some mathematicians may not agree with Feynman in expressing the distance in terms of infinitesimals. The term infinitesimal is a controversial concept and some have explained that this term is dangerous or is no longer used in modern mathematics (Alexander, 2014). They do not agree that infinities or infinitesimals exist in the real world. To the surprise of mathematicians, Robinson (1961) proposes a new way to reintroduce Leibniz’s infinitesimals as a precisely defined mathematical entity. Robinson’s way of using infinitesimals in calculus is known as “non-standard analysis.” There are still some mathematicians criticizing this method of analysis.

2. Integration process:

“This process of adding all these terms together is called integration, and it is the opposite process to differentiation (Feynman et al., 1963, section 8.4 Distance as an integral).”

Feynman’s explanation of integration is rather brief. In essence, he explains that the Δ is replaced by a “d” to remind us that the time is as small as it can be and the addition is written as a sum with a great “s,” ∫ (from the Latin summa). Feynman opines that it is unfortunately just called an integral sign ( ò ) which is only a long S, merely means “the sum of.” However, physics teachers can elaborate that the symbol S may represent the sum of a number of finite quantities, whereas the integral sign ò is always used to represent the summing of an infinite number of “infinitely small quantities.” Alternatively, one may explain geometrically an “integral” as the total area under a curve. Furthermore, we can write the relationship as “distance = ∫ speed (t) dt” or “displacement = ∫ velocity (t) dt.”

Physics teachers should explain why integration is sometimes known as anti-derivative or why would the integral of the velocity bring you back to displacement? We should recall that the area under any curve can be approximated by using a large number of rectangles. If the rectangles are getting thinner and thinner, we can achieve better approximations in determining the correct area. When the limit of rectangles are infinitely thin, we can use the integral symbol as follows: ∫ v(t) dt = ∑_iv(t_i)(t_i+1 − t_i). Importantly, we should remember that the rectangles mainly mean “velocity times time,” which is simply a distance traveled over the little interval, t_i+1 − t_i. By adding together all the distances traveled (or more precisely, displacement) over all those little bits of time, it gives us the net displacement (or change in position).

3. Definite function:

“Every function can be differentiated analytically, i.e., the process can be carried out algebraically, and leads to a definite function (Feynman et al., 1963, section 8.4 Distance as an integral).”

According to Feynman, every function can be differentiated analytically and the process can lead to a definite function. This statement is not true because there are different kinds of functions that cannot be differentiated analytically. First, there are piecewise continuous functions that are not differentiable at points of discontinuity. Second, a function such as sin (1/x) cannot be differentiated at x = 0 because it “oscillates” rapidly and thus, it cannot be defined. Third, a function such as x^1/3 is finite everywhere, but its derivative is infinite at x = 0. In short, Feynman could have said, “most continuous functions can be differentiated analytically.” Nevertheless, there are also exceptions in which non-continuous functions can be differentiated.

Mathematicians or mathematical physicists may find the last paragraph of this section unsatisfactory. Some may add that indefinite integral (ò f(x) dx) is a function, whereas definite integral (ò^b_a f(x) dx) that represents the area under the curve f(x) from x = a to x = b has a definite value. Additionally, Feynman simply mentions that it is not possible to find, analytically, what the integral is for some functions. Therefore, one may clarify that the integral of some functions do not have elementary functions. (An elementary function is a function of one variable which composes a finite number of functions that are algebraic, trigonometric, exponential, logarithmic or constant.) Examples of non-elementary integrals are ò ln(ln x) dx, ò sin x² dx, and ò sin x/x dx.

Questions for discussion:

1. Is it meaningful to define a distance in terms of infinitesimals?

2. How would you explain the integration process geometrically?

3. Can every function be differentiated analytically and resulted in a definite function?

The moral of the lesson: the distance from the origin of an object can be calculated by summing an infinite number of “infinitely thin rectangles” under a curve.

References:

1. Alexander, A. (2014). Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. New York: Farrar, Straus, and Giroux.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Robinson, A. (1961). Non-Standard Analysis. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, ser. A, 64, 432-440.

Section 8–3 Speed as a derivative

(Mathematical notations / Limiting process / Differentiating functions)

In this section, the three interesting points discussed are mathematical notations, limiting process, and differentiation functions.

1. Mathematical notations:

“…The prefix Δ is not a multiplier, any more than sin θ means s i n θ — it simply defines a time increment, and reminds us of its special character (Feynman et al., 1963, section 8.3 Speed as a derivative).”

As a matter of convenience, special notations Δt and Δs have been assigned to replace the quantities ϵ and x respectively. According to Feynman, Δt means “an extra bit of t” and carries an implication that it can be made smaller. Next, he clarifies that the prefix Δ is not a multiplier and it simply means a time increment. In short, Δ is not a factor and it cannot be canceled in the ratio Δs/Δt to give s/t. To be precise, Δt may refer to infinitely small quantities that are smaller than any positive number of the real number system and it does not equal to zero. Moreover, the notation Δt can be replaced by dt, in which Δ and d are known as the “difference symbol” and “differential symbol” respectively.

On the other hand, Feynman elaborates that Δs has an analogous meaning for the distance s. However, physics teachers may explain that the notation Δs means a difference in displacement instead of distance. Furthermore, the notation s is related to the Latin word spatium, which means “a stretch or extent” in space. Importantly, velocity is equal to the limit of Δs/Δt as Δt approaches zero and it can be represented by ds/dt. Historically, Newton introduced the notation ḟ for the derivative of a function f, whereas Leibniz wrote the derivative of a function y with respect to the independent variable x as in dy/dx. One may prefer Leibniz’s notation because it tells us about the independent variable such as x, z, or t.

2. Limiting process:

“This statement is true only if the velocity is not changing during that time interval, and this condition is true only in the limit as Δt goes to 0 (Feynman et al., 1963, section 8.3 Speed as a derivative).”

In a sense, a philosophy of calculus is to make the best approximation. It is not surprising to see Feynman uses the word approximation three times in the paragraph of explaining the limiting process. Firstly, he mentions that to a good “approximation” we have another law, which states that the change in distance of a moving point is the velocity times the time interval, or Δs = vΔt. Secondly, Feynman clarifies that physicists like to write ds = vdt because by dt they mean Δt in circumstances in which it is very small and the expression is valid to a close “approximation.” Thirdly, he adds that if Δt is too long, the velocity might not be constant during the interval, and the “approximation” would become less accurate. Essentially, physicists write v = lim_Δt→0 Δs/Δt = ds/dt based on the “approximation” or limiting process.

A purpose of the limiting process in calculus is to avoid an embarrassing situation (or a definition problem): dy/dx = 0/0. In Principia, Newton (1999) first explains that “[t]hose ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity (p. 442).” However, Feynman did not like the symbol dy/dx. In his own words, “I didn’t like f(x) -- that looked to me like f times x. I also didn’t like dy/dx -- you have a tendency to cancel the d’s -- so I made a different sign, something like an & sign... I thought my symbols were just as good, if not better, than the regular symbols -- it doesn’t make any difference what symbols you use -- but I discovered later that it does make a difference (Feynman, 1997, p. 24).”

3. Differentiating functions:

“This is the fundamental process of calculus, differentiating functions. The process is even more simple than it appears (Feynman et al., 1963, section 8.3 Speed as a derivative).”

Feynman mentions that the terms ds or dt are called differentials, and clarifies that higher power of Δt may be dropped because they approach to 0 when the limit is taken. Simply put, the notation dx means a little bit of x and more precisely, the word differential refers to an infinitesimal (infinitely small) quantity. Interestingly, Bertrand Russell (1992) argues that “infinitesimals as explaining continuity must be regarded as unnecessary, erroneous and self-contradictory (p. 350).” Currently, there is no agreement on the usefulness of the concept of infinitesimal. Gardner writes that “[d]ebate over the infinitesimal versus the limit language goes nowhere because they are two ways of saying the same thing (Thompson & Gardner, 1998, p. 24).

Feynman states the quantity ds/dt as the “derivative of s with respect to t” and adds that the process of differentiating (or finding a derivative) is simpler than it appears. Furthermore, he suggests the rules for differentiating various types of functions can be memorized or can be found in tables. Nevertheless, one may prefer an intuitive or geometric explanation of differentiation. In general, physics teachers can explain the derivation of “power rule” (d/dx [xⁿ] = n x^n-1) involves binomial theorem. Better still, the process of differentiation can be visualized by using an applet that illustrates the slope of a tangent line at a point of a displacement-time graph (or position-time graph).

Questions for discussion:

1. What are the correct meanings of Δ and d in calculus?

2. How would you explain the limiting process?

3. How would you provide an intuitive or geometric explanation of differentiation?

The moral of the lesson: speed is a derivative of distance with respect to time.

References:

1. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Newton, I. (1999/1687). The Principia: Mathematical principles of natural philosophy, translated by I. B. Cohen & A. Whitman. Berkeley: University of California Press.

4. Russell, B. (1992/1903). The Principles of Mathematics. London: Routledge.

5. Thompson, S. P. & Gardner, M. (1998). Calculus made easy. New York: St. Martin’s Press.

Section 8–2 Speed

(Zeno’s paradox / Defining velocity / Measuring speed)

In this section, the three interesting points discussed are Zeno’s paradox, a theoretical definition of velocity, and an empirical definition of speed.

1. Zeno’s paradox:

“…Zeno produced a large number of paradoxes, of which we shall mention one to illustrate his point that there are obvious difficulties in thinking about motion (Feynman et al., 1963, section 8.2 Speed).”

Feynman describes a Zeno’s paradox of motion (Achilles and the tortoise) and explains that a finite amount of time can be divided into an infinite number of pieces just like a finite length of a line can be divided into an infinite number of pieces. Although there is an infinite number of steps to the point at which Achilles reaches the tortoise, it doesn’t mean that they require an infinite amount of time. Mathematicians may explain that this Zeno’s paradox of motion was resolved by Cantor or Cauchy sum. For example, some elaborate that an infinite sum of numbers: 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +... is equal to a finite number, 2. On the other hand, physicists may discuss the concept of Planck length and argue that space is made of finite and discrete units. There is no agreement how the Zeno’s paradox should be resolved.

Alternatively, one may prefer to discuss Zeno’s Paradox of the Arrow: 1. The arrow always occupies a portion of space that is equal to its own length. 2. At any instant of its flight, the arrow can be located in a place having the same length. 3. One may conclude that at every instance of the flight, the arrow is at rest. An instant is a minimal and indivisible element of time. According to Aristotle, the paradox assumes that time is composed of “nows” (or indivisible instants). However, this paradox is relevant to the concept of instantaneous velocity that is useful and important in physics: it can be defined as the limit of the sequence of x’s average velocities for increasingly small intervals of time containing t.

2. Defining velocity:

“…Calculus was invented in order to describe motion, and its first application was to the problem of defining what is meant by going ‘60 miles an hour’ (Feynman et al., 1963, section 8.2 Speed).”

Feynman discusses a cop’s definition of velocity and problems of defining the same velocity. For example, a lady may argue that if a car kept going at the same velocity say “60 miles an hour,” she would run into a wall at the end of the street! However, a theoretical definition of velocity involves the idea of an infinitesimal distance and infinitesimal time, as well as takes a limit of the distance traveled divided by the time required, as the time taken gets smaller and smaller, ad infinitum. To be precise, in a short time, ϵ, when the car or any object moves a short distance x, then the velocity, v, is defined as v = x/ϵ, an approximation that becomes better and better as the ϵ is taken smaller and smaller. This is a concept of instantaneous velocity that is based on a branch of mathematics, called the differential calculus.

Feynman did not explicitly state a definition of velocity as the rate of change of displacement of an object per unit time nor specify the velocity is with respect to an inertial frame of reference. Interestingly, he got into trouble through his discussion of the cop’s definition of velocity. In his words, “a few years after I gave some lectures for the freshmen at Caltech (which were published as the Feynman Lectures on Physics), I received a long letter from a feminist group. I was accused of being anti-woman because of two stories: the first was a discussion of the subtleties of velocity and involved a woman driver being stopped by a cop. There's a discussion about how fast she was going, and I had her raise valid objections to the cop’s definitions of velocity. The letter said I was making the woman look stupid (Feynman, 1988, p. 72).”

3. Measuring speed:

“Many physicists think that measurement is the only definition of anything. Obviously, then, we should use the instrument that measures the speed — the speedometer (Feynman et al., 1963, section 8.2 Speed).”

Some physicists advocate the importance of empirical definitions of physics concepts. As an example, some may define speed as measured by using a speedometer. However, Feynman argues that the measuring instrument that determines the speed may not be under ideal working conditions. One may deduce that “the speedometer isn’t working right,” or “the speedometer is broken.” Importantly, the speedometer is based on a theoretical definition of velocity or instantaneous speed. In essence, physicists’ measurement of the speed of an object is related to the theoretical definition of velocity.

Strictly speaking, an empirical definition of speed as “measured by a speedometer” may not be accurate because of a change in wheel size or the car’s transmission/drive ratios. Currently, physicists may prefer to use Global Positional System (GPS) speedometers as positional tracking systems that can be more accurate. These speedometers are dependent on physicists’ ideas of space-time and how they synchronize the time at different locations on Earth. In short, the speed measured is also with respect to the earth’s frame of reference. However, the accuracies of GPS speedometers are subjected to satellites errors, atmospheric effects, and relativistic effects.

Questions for discussion:

1. How would you resolve Zeno’s paradox of motion?

2. How would you provide a theoretical definition of speed?

3. How would you provide an empirical definition of speed?

The moral of the lesson: we need a rigorous theoretical definition of speed as well as an accurate empirical definition of speed.

References:

1. Feynman, R. P. (1988). What Do You Care What Other People Think? New York: W W Norton

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.