2.3

Acceleration

The velocity of a moving object may change in a number of ways. For example, it may increase, as it does when the driver of a car steps on the gas pedal to pass the car ahead. Or it may decrease, as it does when the driver applies the brakes to stop at a red light. In either case, the change in velocity may occur over a short or a long time interval. To describe how the velocity of an object changes during a given time interval, we now introduce the new idea of acceleration. This idea depends on two concepts that we have previously encountered, velocity and time. Specifically, the notion of acceleration emerges when the change in the velocity is combined with the time during which the change occurs.
The meaning of average acceleration can be illustrated by considering a plane during takeoff. Figure 2-4 focuses attention on how the plane’s velocity changes along the runway. During an elapsed time interval , the velocity changes from an initial value of to a final velocity of . The change in the plane’s velocity is its final velocity minus its initial velocity, so that . The average acceleration is defined in the following manner, to provide a measure of how much the velocity changes per unit of elapsed time.

Figure 2-4 During takeoff, the plane accelerates from an initial velocity to a final velocity during the time interval .

| Definition of Average Acceleration

| (2.4)

SI Unit of Average Acceleration: meter per second squared (m/s2)

The average acceleration is a vector that points in the same direction as , the change in the velocity. Following the usual custom, plus and minus signs indicate the two possible directions for the acceleration vector when the motion is along a straight line.

A skier accelerates out of the starting gate. (Agence Zoom/Getty Sport/Getty Images Sport Services)

We are often interested in an object’s acceleration at a particular instant of time. The instantaneous acceleration can be defined by analogy with the procedure used in Section 2.2 for instantaneous velocity:

| (2.5)

Equation 2.5 indicates that the instantaneous acceleration is a limiting case of the average acceleration. When the time interval for measuring the acceleration becomes extremely small (approaching zero in the limit), the average acceleration and the instantaneous acceleration become equal. Moreover, in many situations the acceleration is constant, so the acceleration has the same value at any instant of time. In the future, we will use the word acceleration to mean “instantaneous acceleration.” Example 3 deals with the acceleration of a plane during takeoff.

| Example 3 | | | Acceleration and Increasing Velocity

Suppose the plane in Figure 2-4 starts from rest ( = 0 m/s) when t0 = 0 s. The plane accelerates down the runway and at t = 29 s attains a velocity of = +260 km/h, where the plus sign indicates that the velocity points to the right. Determine the average acceleration of the plane.

Problem solving insight The change in any variable is the final value minus the initial value: for example, the change in velocity is , and the change in time is ().

Reasoning The average acceleration of the plane is defined as the change in its velocity divided by the elapsed time. The change in the plane’s velocity is its final velocity minus its initial velocity , or . The elapsed time is the final time t minus the initial time t0, or .
Solution The average acceleration is expressed by Equation 2.4 as

The average acceleration calculated in Example 3 is read as “nine kilometers per hour per second.” Assuming the acceleration of the plane is constant, a value of means the velocity changes by +9.0 km/h during each second of the motion. During the first second, the velocity increases from 0 to 9.0 km/h; during the next second, the velocity increases by another 9.0 km/h to 18 km/h, and so on. Figure 2-5 illustrates how the velocity changes during the first two seconds. By the end of the 29th second, the velocity is 260 km/h.

Figure 2-5 An acceleration of means that the velocity of the plane changes by during each second of the motion. The “+” direction for and is to the right.

It is customary to express the units for acceleration solely in terms of SI units. One way to obtain SI units for the acceleration in Example 3 is to convert the velocity units from km/h to m/s:

The average acceleration then becomes

where we have used . An acceleration of is read as “2.5 meters per second per second” (or “2.5 meters per second squared”) and means that the velocity changes by 2.5 m/s during each second of the motion.
Example 4 deals with a case where the motion becomes slower as time passes.

| Example 4 | | | Acceleration and Decreasing Velocity

A drag racer crosses the finish line, and the driver deploys a parachute and applies the brakes to slow down, as Figure 2-6 illustrates. The driver begins slowing down when and the car’s velocity is . When , the velocity has been reduced to . What is the average acceleration of the dragster?

Figure 2-6 (a) To slow down, a drag racer deploys a parachute and applies the brakes. (b) The velocity of the car is decreasing, giving rise to an average acceleration that points opposite to the velocity. (a, Dave Kommel/AutoImagery)

Reasoning The average acceleration of an object is always specified as its change in velocity, , divided by the elapsed time, . This is true whether the final velocity is less than the initial velocity or greater than the initial velocity.
Solution The average acceleration is, according to Equation 2.4,

Problem solving insight

Figure 2-7 shows how the velocity of the dragster changes during the braking, assuming that the acceleration is constant throughout the motion. The acceleration calculated in Example 4 is negative, indicating that the acceleration points to the left in the drawing. As a result, the acceleration and the velocity point in opposite directions. Whenever the acceleration and velocity vectors have opposite directions, the object slows down and is said to be “decelerating.” In contrast, the acceleration and velocity vectors in Figure 2-5 point in the same direction, and the object speeds up.

Figure 2-7 Here, an acceleration of −5.0 m/s2 means the velocity decreases by 5.0 m/s during each second of elapsed time.

Check Your Understanding 2

Two cars are moving on straight sections of a highway. The acceleration of the first car is greater than the acceleration of the second car, and both accelerations have the same direction. Which one of the following is true? (a) The velocity of the first car is always greater than the velocity of the second car. (b) The velocity of the second car is always greater than the velocity of the first car. (c) In the same time interval, the velocity of the first car changes by a greater amount than the velocity of the second car does. (d) In the same time interval, the velocity of the second car changes by a greater amount than the velocity of the first car does. (The answer is given at the end of the book.)
Background: Velocity and acceleration are the focus of this question. They are not the same concept.
For similar questions (including calculational counterparts), consult Self-Assessment Test 2.1, which is described next.

Test your understanding of the key concepts in Sections 2.1–2.3:

· | Displacement
· | Velocity
· | Acceleration

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