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.