Section 9–3 Components of velocity, acceleration, and force

(Components of velocity / Components of acceleration / Components of a force)

In this section, the three interesting concepts are components of velocity, components of acceleration, and components of a force.

1. Components of velocity:

“…we have resolved the velocity into components by telling how fast the object is moving in the x-direction, the y-direction, and the z-direction (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force)

Feynman mentions that the velocity of an object is completely specified if we give the numerical values of its three perpendicular components: v_x = dx/dt, v_y = dy/dt, v_z = dz/dt. Furthermore, the magnitude of the velocity of the object can be calculated by using the equation, ds/dt = √(v_x²+ v_y² + v_z²). Essentially, the components of velocity refer to the speed of the object in the x-direction, the y-direction, and the z-direction. We can demonstrate these components of velocity by using a light source or projector. If we shine light vertically downward on a moving object, we can observe a shadow (or projection) moves in a specific direction. Physics teachers may explain that a component of velocity is projected onto the x-axis or y-axis depending on the direction of light rays.

There are gaps in Feynman’s explanation of components of velocity because this is a relatively easy topic. In Tips on Physics, Feynman adds that “the velocity in terms of x, y, and z components is very easy, because, for example, the rate of change of the x component of the position is equal to the x component of velocity, and so on. This is simply because the derivative is really a difference, and since the components of a difference vector equal the differences of the corresponding components (Feynman et al., 2006, p. 30).” In other words, the derivative of a position vector is related to a difference in positions of an object. Mathematically, the components (or shadows of an object) of a vector in the three-dimensional world also obey Newton’s laws of motion.

2. Components of acceleration:

The change in the component of the velocity in the x-direction in a time Δt is Δv_x = a_xΔt, where a_x is what we call the x-component of the acceleration (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force)

The action of a force can cause the velocity of an object changes to another direction and a different magnitude. Feynman explains that this apparently complex situation can be simply analyzed by evaluating the changes in the x-, y-, and z-components of velocity. Mathematically, the change in the component of the velocity in the x-direction in a short time Δt is Δv_x = a_xΔt, in which a_x is the x-component of the acceleration. Without loss of generality, we have Δv_y = a_yΔt and Δv_z = a_zΔt. Essentially, we can resolve the displacement, velocity, and acceleration of an object into components by projecting a line segment to represent these quantities.

In The Evolution of Physics, Einstein and Infeld (1938) write that “[b]y following the right clue, we achieve a deeper understanding of the problem of motion. The connection between force and the change of velocity and not, as we should think according to our intuition, the connection between force and the velocity itself is the basis of classical mechanics as formulated by Newton (p. 10).” In short, force is connected to a change in velocity instead of simply velocity. We should recall Feynman’s explanation that “the derivative is really a difference (Feynman et al., 2006, p. 30).” Thus, one may explain the connection by using the concept of “change in velocity” instead of only acceleration.

3. Components of a force:

“If we know the forces on an object and resolve them into x-, y-, and z-components, then we can find the motion of the object from these equations (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).”

Feynman suggests that there are really “three” laws in the sense that the component of the force in the x-, y-, or z-direction is equal to the mass of an object times the rate of change of the corresponding component of velocity: F_x = m(dv_x/dt) = m(d²x/dt²) = ma_x, F_y = m(dv_y/dt) = m(d²y/dt²) = ma_y, F_z = m(dv_z/dt) = m(d²z/dt²) = ma_z. One may infer that Newton’s Second Law can also be represented by infinite possible combinations of x-, y-, or z-direction and hence there is an infinite number of laws governing the force in various directions. However, it is possible to simplify the motion of an object by using only two equations or even one equation depending on how we choose the x-, y-, or z-direction. Thus, Feynman does not need to identify each equation as a theoretical law.

Feynman states that motions in the x-, y-, and z-direction are independent if the forces are not connected. Historically, in his investigations of motion, Galileo is the first person to conceptualize the forces acting upon objects could be resolved into independent components. In Dialogues Concerning Two New Sciences, he writes that “the resulting motion which I call projection is compounded of one which is uniform and horizontal and of another which is vertical and naturally accelerated (Galilei, 1638, p. 244).” Galileo’s insights are remarkable because the ideal motion of projectile motion could not be directly observed due to the presence of air resistance. Importantly, physicists have assumed Euclidean geometry of space in the analysis of motions.

Questions for discussion:

1. Why are we allowed to resolve velocity into perpendicular components?

2. Why is a force connected to a change in velocity instead of velocity?

3. Why are we allowed to resolve forces into perpendicular components?

The moral of the lesson: force is connected to a change in velocity instead of velocity.

References:

1. Einstein, A. & Leopold, I. (1938). The Evolution of Physics. New York: Simon & Schuster.

2. Feynman, R. P., Gottlieb, M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

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. (1638/1914). Dialogues Concerning Two New Sciences. New York: Dover.

Section 9–2 Speed and velocity

(Redefining speed & velocity / Comparing speed & velocity / Formulating velocity)

In this section, the three interesting concepts are redefining speed and velocity, comparing speed and velocity, and formulating velocity.

1. Redefining speed and velocity:

“Ordinarily we think of speed and velocity as being the same, and in ordinary language they are the same (Feynman et al., 1963, section 9–2 Speed and velocity).”

In a lecture on quantum electrodynamics, Feynman (1985) explains that physicists use ordinary words such as work, action, energy, or light, in a funny way. Physicists also redefine speed and velocity that have the same meaning in daily life. In Regulae solvendi sophismata, Heytesbury defines the “velocity at any instant in non-uniform motion as the ratio of the distance traveled to the time that would have elapsed if the motion had been uniform at that velocity (Weinberg, 2015, p. 138).” Weinberg (2015) mentions that this definition is circular and hence useless. Grant (1996) explains that it defines “instantaneous velocity” by a uniform speed that is equal to the instantaneous velocity (it is yet to be defined). However, Heytesbury also derives the mean speed theorem that may be expressed as s = ½(v_i + v_f)t.

In 1928, Einstein posed the following questions to Jean Piaget: “Is our intuitive grasp of time primitive or derived? Is it identical with our intuitive grasp of velocity? (Piaget, 1969, p. xiii).” Einstein wanted to know whether children’s understanding of these concepts was intuitive or derived, and how their understanding of one concept influenced subsequent understanding of the other. Based on his findings, Piaget (1972) explains that “[t]he relationship v = d/t implies that v is a relationship and that both d and t are straightforward intuitions. The truth, however, is that some intuitions of speed, such as those of overtaking, actually precede those of time (p. 78).” In other words, children do not necessarily think of velocity in terms of the distance-time relationship and their concept of time could be derived from velocity.

2. Comparing speed and velocity:

“We carefully distinguish velocity, which has both magnitude and direction, from speed, which we choose to mean the magnitude of the velocity, but which does not include the direction (Feynman et al., 1963, section 9–2 Speed and velocity).”

Dictionary definitions of speed and velocity have essentially the same meaning. Currently, we can compare the concepts of speed and velocity from the perspectives of theoretical definition, classification, and equation. Firstly, speed is commonly defined as the rate of change of distance traveled by an object and velocity is the rate of change of displacement of an object. Secondly, the speed of an object can be classified as a scalar quantity and velocity is a vector quantity. Thirdly, speed can be mathematically represented by v = d/t which means a ratio of distance moved (d) over an interval of time (t) whereas velocity can be represented in terms of three components: v = v_x i + v_y j + v_z k. These three differences can be simply explained by the fact that speed is directionless in contrast to velocity that has a specific direction.

Some may prefer Einstein and Infeld’s (1938) comparison of speed and velocity: “consider two spheres moving in different directions on a smooth table. So as to have a definite picture, we may assume the two directions perpendicular to each other. Since there are no external forces acting, the motions are perfectly uniform. Suppose, further, that the speeds are equal, that is, both cover the same distance in the same interval of time. But is it correct to say that the two spheres have the same velocity? The answer can be yes or no! If the speedometers of two cars both show forty miles per hour, it is usual to say that they have the same speed or velocity, no matter in which direction they are traveling. But science must create its own language, its own concepts, for its own use. Scientific concepts often begin with those used in ordinary language for the affairs, of everyday life, but they develop quite differently. They are transformed and lose the ambiguity associated with them in ordinary language, gaining in rigorousness so that they may be applied to scientific thought (p. 12).”

3. Formulating velocity:

“We can formulate this more precisely by describing how the x-, y-, and z-coordinates of an object change with time (Feynman et al., 1963, section 9–2 Speed and velocity).”

In general, the motion of a particle in a specific direction can be resolved into three components that are independent of each other. Therefore, the position of the particle can be mathematically represented by three independent equations in terms of x, y, and z. Feynman explains that we can formulate the particle’s motion by describing how the x-, y-, and z-coordinates change with time. In a short interval of time Δt, we can assume the particle moves in a straight line and the total distance moved (Δs) can be resolved as a certain distance Δx in the x-direction, Δy in the y-direction, and Δz in the z-direction. Mathematically, the displacement Δx is equal to the x-component of the velocity times Δt, that is, Δx = v_xΔt. Similarly, we have Δy = v_yΔt and Δz = v_zΔt.

This concept of velocity is formulated based on the assumption of Euclidean geometry. In Feynman’s Tips on physics, he elaborates that “[i]n this case, where A is position, its derivative is a velocity vector; the velocity vector is in a direction tangent to the curve, because that's the direction of the displacements; its magnitude you can’t get by looking at this picture, because it depends on how fast the thing is going along the curve. The magnitude of the velocity vector is the speed; it tells you how far the thing moves per unit time. So, that's a definition of the velocity vector: it’s tangent to the path, and its magnitude is equal to the speed of motion on the path (Feynman, 2006, p. 29).” Because velocity is defined as a vector, it also needs to follow mathematical rules with regard to vector differentiation.

Questions for discussion:

1. How would you redefine the speed and velocity of an object?

2. How would you compare the differences between speed and velocity?

3. How would you formulate velocity in terms of vector quantities?

The moral of the lesson: physicists redefine ordinary words such as speed and velocity that have the same meaning in daily life.

References:

1. Einstein, A. & Leopold, I. (1938). The Evolution of Physics. New York: Simon & Schuster.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. Feynman, R. P., Gottlieb, M. A., & Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

4. 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.

5. Grant, E. (1996). The Foundations of Modern Science in the Middle Ages: Their Religious, Institutional and Intellectual Contexts. Cambridge: Cambridge University Press.

6. Piaget, J. (1969). The Child’s Conception of Time. New York: Basic Books.

7. Piaget, J. (1972). Psychology and Epistemology: Towards a Theory of Knowledge. Middlesex: Penguin.

8. Weinberg, S. (2015). To Explain the World: The Discovery of Modern Science. London: Allen Lane.

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.