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.