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.