In 《 SECTION 1.5 we defined the average acceleration $\vec{a}{}_{\text{avg}}$ of a moving object to be

From its definition, we see that $\vec{a}$ points in the same direction as $\text{Δ}\vec{v}$, the change of velocity. As an object moves, its velocity vector can change in two possible ways:

1. The magnitude of $\vec{v}$ can change, indicating a change in speed, or

2. The direction of $\vec{v}$ can change, indicating that the object has changed direction.

Chapters 1 and 2 considered only the acceleration of changing speed. The acceleration of changing direction can be determined by finding the direction of $\text{Δ}\vec{v}$. If an object changes from velocity $\vec{v}{}_{\text{a}}$ to velocity $\vec{v}{}_{\text{b}}$, its change of velocity $\text{Δ}\vec{v}=\vec{v}{}_{\text{b}}-\vec{v}{}_{\text{a}}$ can be written as $\vec{v}{}_{\text{b}}=\vec{v}{}_{\text{a}}+\text{Δ}\vec{v}$. Thus $\text{Δ}\vec{v}$ is the vector that must be added to $\vec{v}{}_{\text{a}}$ to get $\vec{v}{}_{\text{b}}$. Tactics Box 4.1 shows how to use graphical vector addition to find the acceleration vector.

TACTICS BOX 4.1 Finding the acceleration vector

To find the acceleration between velocity ${\overrightarrow{v}}_{\text{a}}$ and velocity ${\overrightarrow{v}}_{\text{b}}$:

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1 Draw velocity vectors ${\overrightarrow{v}}_{\text{a}}$ and ${\overrightarrow{v}}_{\text{b}}$ with their tails together.

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2 Draw the vector from the tip of ${\overrightarrow{v}}_{\text{a}}$ to the tip of ${\overrightarrow{v}}_{\text{b}}$. This is $\Delta \overrightarrow{v}$ because ${\overrightarrow{v}}_{\text{b}}={\overrightarrow{v}}_{\text{a}}+\Delta \overrightarrow{v}$.

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3 Return to the original motion diagram. Draw a vector at the middle dot in the direction of $\Delta \overrightarrow{v}$; label it $\overrightarrow{a}$. This is the average acceleration between ${\overrightarrow{v}}_{\text{a}}$ and ${\overrightarrow{v}}_{\text{b}}$.

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Workbook Exercises 1–4

Our everyday use of the word “accelerate” means “speed up.” The mathematical definition of acceleration—the rate of change of velocity—also includes slowing down, as you learned in Chapter 2, as well as changing direction. All these are motions that change the velocity.

EXAMPLE 4.2 Through the valley

A ball rolls down a long hill, through the valley, and back up the other side. Draw a complete motion diagram of the ball.

MODEL Model the ball as a particle.

VISUALIZE FIGURE 4.4 is the motion diagram. Where the particle moves along a straight line, it speeds up if $\vec{a}$ and $\vec{v}$ point in the same direction and slows down if $\vec{a}$ and $\vec{v}$ point in opposite directions. This important idea was the basis for the one-dimensional kinematics we developed in Chapter 2. When the direction of $\vec{v}$ changes, as it does when the ball goes through the valley, we need to use Tactics Box 4.1 to find the direction of $\text{Δ}\vec{v}$ and thus of $\vec{a}$. The procedure is shown at one point in the motion diagram. Notice that $\vec{a}$ is perpendicular to the trajectory at the bottom point where only the direction, not the speed, is changing. We’ll return to this idea when we discuss circular motion.

FIGURE 4.4 The motion diagram of the ball of Example 4.2.

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FIGURE 4.5 shows that an object’s acceleration vector can be decomposed into a component parallel to the velocity—that is, parallel to the direction of motion—and a component perpendicular to the velocity. ${\overrightarrow{a}}_{\parallel }$ is the piece of the acceleration that causes the object to change speed, speeding up if ${\overrightarrow{a}}_{\parallel }$ points in the same direction as $\vec{v}$, slowing down if they point in opposite directions. $\vec{a}{}_{\perp }$ is the piece of the acceleration that causes the object to change direction. An object changing direction always has a component of acceleration perpendicular to the direction of motion.

FIGURE 4.5 Analyzing the acceleration vector.

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Looking back at Example 4.2, we see that $\vec{a}$ is parallel to $\vec{v}$ on the straight portions of the hill where only speed is changing. At the very bottom, where the ball’s direction is changing but not its speed, $\vec{a}$ is perpendicular to $\vec{v}$. The acceleration is angled with respect to velocity—having both parallel and perpendicular components—at those points where both speed and direction are changing.