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.