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.