We need one more major concept to describe one-dimensional motion: acceleration. Acceleration, as we noted in Chapter 1, is a rather abstract concept. Nonetheless, acceleration is the linchpin of mechanics. We will see very shortly that Newton’s laws relate the acceleration of an object to the forces that are exerted on it.

Let’s conduct a race between a Volkswagen Beetle and a Porsche to see which can reach a speed of 30 m/s $(\approx 60\text{mph})$ in the shortest time. Both cars are equipped with computers that will record the speedometer reading 10 times each second. This gives a nearly continuous record of the instantaneous velocity of each car. TABLE 2.1 shows some of the data. The velocity-versus-time graphs, based on these data, are shown in FIGURE 2.19 .

FIGURE 2.19 Velocity-versus-time graphs for the Porsche and the VW Beetle.

How can we describe the difference in performance of the two cars? It is not that one has a different velocity from the other; both achieve every velocity between 0 and 30 m/s. The distinction is how long it took each to change its velocity from 0 to 30 m/s. The Porsche changed velocity quickly, in 6.0 s, while the VW needed 15 s to make the same velocity change. Because the Porsche had a velocity change $\text{Δ}{v}_s=30\text{m}/\text{s}$ during a time interval $\text{Δ}t=6.0$ s, the rate at which its velocity changed was

Notice the units. They are units of “velocity per second.” A rate of velocity change of 5.0 “meters per second per second” means that the velocity increases by $5.0\text{m}/\text{s}$ during the first second, by another $5.0\text{m}/\text{s}$ during the next second, and so on. In fact, the velocity will increase by $5.0\text{m}/\text{s}$ during any second in which it is changing at the rate of $5.0(\text{m}/\text{s})/\text{s}$.

Chapter 1 introduced acceleration as “the rate of change of velocity.” That is, acceleration measures how quickly or slowly an object’s velocity changes. In parallel with our treatment of velocity, let’s define the average acceleration $a_{\text{avg}}$ during the time interval $\text{Δ}t$ to be

Equations 2.15 and 2.16 show that the Porsche had the rather large acceleration of 5.0 $(\text{m}\text{/}\text{s})\text{/}\text{s}$.

Because $\text{Δ}{v}_s$ and $\text{Δ}t$ are the “rise” and “run” of a velocity-versus-time graph, we see that $a_{\text{avg}}$ can be interpreted graphically as the slope of a straight-line velocity-versus-time graph. In other words,

Figure 2.19 uses this idea to show that the VW’s average acceleration is

This is less than the acceleration of the Porsche, as expected.

An object whose velocity-versus-time graph is a straight-line graph has a steady and unchanging acceleration. There’s no need to specify “average” if the acceleration is constant, so we’ll use the symbol $a_s$ as we discuss motion along the s-axis with constant acceleration.