Learning Objectives
By the end of this section, you will be able to do the following:
- Define and distinguish between instantaneous acceleration and average acceleration
- Calculate acceleration given initial time, initial velocity, final time, and final velocity
The information presented in this section supports the following AP® learning objectives and science practices:
- 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)
- 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1)
In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time. The formal definition of acceleration is consistent with these notions but more inclusive.
Average Acceleration
Average Acceleration is the rate at which velocity changes,
where is average acceleration, is velocity, and is time. The bar over the means average acceleration.
Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second.
Recall that velocity is a vector—it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both.
Acceleration as a Vector
Acceleration is a vector in the same direction as the change in velocity, Since velocity is a vector, it can change either in magnitude or in direction. Acceleration is therefore a change in either speed or direction, or both.
Keep in mind that, although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object's acceleration is in the same direction of its motion, the object will speed up. However, when an object's acceleration is opposite to the direction of its motion, the object will slow down. Speeding up and slowing down should not be confused with a positive and negative acceleration. The next two examples should help to make this distinction clear.
Making Connections: Car Motion
Consider the acceleration and velocity of each car in terms of its direction of travel.
Because the positive direction is considered to the right of the paper, Car A is moving with a positive velocity. Because it is speeding up while moving with a positive velocity, its acceleration is also considered positive.
Because the positive direction is considered to the right of the paper, Car B is also moving with a positive velocity. However, because it is slowing down while moving with a positive velocity, its acceleration is considered negative. This can be viewed in a mathematical manner as well. If the car was originally moving with a velocity of +25 m/s, it is finishing with a speed less than that, like +5 m/s. Because the change in velocity is negative, the acceleration will be as well.
Because the positive direction is considered to the right of the paper, Car C is moving with a positive velocity. Because all arrows are of the same length, this car is not changing its speed. As a result, its change in velocity is zero, and its acceleration must be zero as well.
Because the car is moving opposite to the positive direction, Car D is moving with a negative velocity. Because it is speeding up while moving in a negative direction, its acceleration is negative as well.
Because it is moving opposite to the positive direction, Car E is moving with a negative velocity as well. However, because it is slowing down while moving in a negative direction, its acceleration is actually positive. As in example B, this may be more easily understood in a mathematical sense. The car is originally moving with a large negative velocity (−25 m/s) but slows to a final velocity that is less negative (−5 m/s). This change in velocity, from −25 m/s to −5 m/s, is actually a positive change: of 20 m/s. Because the change in velocity is positive, the acceleration must also be positive.
Making Connections: Illustrative Example
The three graphs below are labeled A, B, and C. Each one represents the position of a moving object plotted against time.
As we did in the previous example, let's consider the acceleration and velocity of each object in terms of its direction of travel.
Object A is continually increasing its position in the positive direction. As a result, its velocity is considered positive.
During the first portion of time (shaded grey), the position of the object does not change much, resulting in a small positive velocity. During a later portion of time (shaded green), the position of the object changes more, resulting in a larger positive velocity. Because this positive velocity is increasing over time, the acceleration of the object is considered positive.
As in case A, Object B is continually increasing its position in the positive direction. As a result, its velocity is considered positive.
During the first portion of time (shaded grey), the position of the object changes a large amount, resulting in a large positive velocity. During a later portion of time (shaded green), the position of the object does not change as much, resulting in a smaller positive velocity. Because this positive velocity is decreasing over time, the acceleration of the object is considered negative.
Object C is continually decreasing its position in the positive direction. As a result, its velocity is considered negative.
During the first portion of time (shaded grey), the position of the object does not change a large amount, resulting in a small negative velocity. During a later portion of time (shaded green), the position of the object changes a much larger amount, resulting in a larger negative velocity. Because the velocity of the object is becoming more negative during the time period, the change in velocity is negative. As a result, the object experiences a negative acceleration.
Example 2.1 Calculating Acceleration: A Racehorse Leaves the Gate
A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration?
Strategy
First, we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity.
We can solve this problem by identifying and from the given information and then calculating the average acceleration directly from the equation .
Solution
1. Identify the knowns. (the minus sign indicates direction toward the west),
2. Find the change in velocity. Since the horse is going from zero to its change in velocity equals its final velocity:
3. Plug in the known values ( and ) and solve for the unknown .
Discussion
The minus sign for acceleration indicates that acceleration is toward the west. An acceleration of due west means that the horse increases its velocity by 8.33 m/s due west each second; that is, 8.33 meters per second per second, which we write as This is truly an average acceleration, because the ride is not smooth. We shall see later that an acceleration of this magnitude would require the rider to hang on with a force nearly equal to his weight.