1.5 Linear Acceleration

Position, time, and velocity are important concepts, and at first glance they might appear to be sufficient to describe motion. But that is not the case. Sometimes an object’s velocity is constant, as it was in Figure 1.10. More often, an object’s velocity changes as it moves, as in Figures 1.11 and 1.12. We need one more motion concept to describe a change in the velocity.

Because velocity is a vector, it can change in two possible ways:

The magnitude can change, indicating a change in speed; or
The direction can change, indicating that the object has changed direction.

We will concentrate for now on the first case, a change in speed. The car accelerating up a hill in Figure 1.11 was an example in which the magnitude of the velocity vector changed but not the direction. We’ll return to the second case in Chapter 4.

When we wanted to measure changes in position, the ratio $Δ \vec{r} / Δ t$ was useful. This ratio is the rate of change of position. By analogy, consider an object whose velocity changes from $\vec{v}_{a}$ to $\vec{v}_{b}$ during the time interval $Δ t$ . Just as $Δ \vec{r} = \vec{r}_{b} - \vec{r}_{a}$ is the change of position, the quantity $Δ \vec{v} = \vec{v}_{b} - \vec{v}_{a}$ is the change of velocity. The ratio $Δ \vec{v}^{} / Δ t$ is then the rate of change of velocity. It has a large magnitude for objects that speed up quickly and a small magnitude for objects that speed up slowly.

The ratio $Δ \vec{v}^{} / Δ t$ is called the average acceleration, and its symbol is $\vec{a}_{avg}$ . The average acceleration of an object during the time interval $Δ t$ , in which the object’s velocity changes by $Δ \vec{v},$ is the vector

\vec{a}_{avg} = \frac{Δ \vec{v}}{Δ t}

(1.7)

The average acceleration vector points in the same direction as the vector $Δ \vec{v}$ .

A photo of an accelerating Audi T T. — The Audi TT accelerates from 0 to 60 mph in 6 s.

Acceleration is a fairly abstract concept. Yet it is essential to develop a good intuition about acceleration because it will be a key concept for understanding why objects move as they do. Motion diagrams will be an important tool for developing that intuition.

NOTE As we did with velocity, we will drop the subscript and refer to the average acceleration as simply $\vec{a}$ . This is adequate for visualization purposes, but not the final word. We will refine the definition of acceleration in Chapter 2.