### Defining Acceleration

Throughout this chapter we will use the following terms: *time*, *displacement*, *velocity*, and *acceleration*. Recall that each of these terms has a designated variable and SI unit of measurement as follows:

- Time:
*t*, measured in seconds (s)
- Displacement: Δ
*d*, measured in meters (m)
- Velocity:
*v*, measured in meters per second (m/s)
- Acceleration:
*a*, measured in meters per second per second (m/s^{2}, also called meters per second squared)
- Also note the following:
- Δ means
*change in*
- The subscript 0 refers to an initial value; sometimes subscript i is instead used to refer to initial value.
- The subscript f refers to final value
- A bar over a symbol, such as $\overline{a}$, means
*average*

Acceleration is the change in velocity divided by a period of time during which the change occurs. The SI units of velocity are m/s and the SI units for time are s, so the SI units for acceleration are m/s^{2}. Average acceleration is given by

$$\overline{a}=\frac{\Delta v}{\Delta t}=\frac{{v}_{\text{f}}-{v}_{0}}{{t}_{\text{f}}-{t}_{0}}\text{.}$$

Average acceleration is distinguished from instantaneous acceleration, which is acceleration at a specific instant in time. The magnitude of acceleration is often not constant over time. For example, runners in a race accelerate at a greater rate in the first second of a race than during the following seconds. You do not need to know all the instantaneous accelerations at all times to calculate average acceleration. All you need to know is the change in velocity (i.e., the final velocity minus the initial velocity) and the change in time (i.e., the final time minus the initial time), as shown in the formula. Note that the average acceleration can be positive, negative, or zero. A negative acceleration is simply an acceleration in the negative direction.

Keep in mind that although acceleration points in the same direction as the *change* in velocity, it is not always in the direction of the velocity itself. When an object slows down, its acceleration is opposite to the direction of its velocity. In everyday language, this is called deceleration; but in physics, it is acceleration—whose direction happens to be opposite that of the velocity. For now, let us assume that motion to the right along the *x*-axis is *positive* and motion to the left is *negative*.

Figure 3.2 shows a car with positive acceleration in (a) and negative acceleration in (b). The arrows represent vectors showing both direction and magnitude of velocity and acceleration.

Velocity and acceleration are both vector quantities. Recall that vectors have both magnitude and direction. An object traveling at a constant velocity—therefore having no acceleration—does accelerate if it changes direction. So, turning the steering wheel of a moving car makes the car accelerate because the velocity changes direction.

### Virtual Physics

#### The Moving Man

With this animation in Figure 3.3, you can produce both variations of acceleration and velocity shown in Figure 3.2, plus a few more variations. Vary the velocity and acceleration by sliding the red and green markers along the scales. Keeping the velocity marker near zero will make the effect of acceleration more obvious. Try changing acceleration from positive to negative while the man is moving. We will come back to this animation and look at the *Charts* view when we study graphical representation of motion.

Grasp Check

Which part, (a) or (b), is represented when the velocity vector is on the positive side of the scale and the acceleration vector is set on the negative side of the scale? What does the car’s motion look like for the given scenario?

- Part (a). The car is slowing down because the acceleration and the velocity vectors are acting in the opposite direction.
- Part (a). The car is speeding up because the acceleration and the velocity vectors are acting in the same direction.
- Part (b). The car is slowing down because the acceleration and velocity vectors are acting in the opposite directions.
- Part (b). The car is speeding up because the acceleration and the velocity vectors are acting in the same direction.