2.4

Equations of Kinematics for Constant Acceleration

Concepts At A Glance It is now possible to describe the motion of an object traveling with a constant acceleration along a straight line. To do so, we will use a set of equations known as the equations of kinematics for constant acceleration. The equations entail no new concepts, because they will be obtained by combining the familiar ideas of displacement, velocity, and acceleration, as the Concepts-at-a-Glance chart* in Figure 2-8 shows. However, they will provide a very convenient way to determine certain aspects of the motion, such as the final position and velocity of a moving object.

Figure 2-8 CONCEPTS AT A GLANCE The equations of kinematics for constant acceleration are obtained by combining the concepts of displacement, velocity, and acceleration. Whenever this runner maintained a constant acceleration during the race, the equations of kinematics could have been used to calculate his displacement and velocity as a function of time. (Stone/Getty Images)

In discussing the equations of kinematics, it will be convenient to assume that the object is located at the origin when . With this assumption, the displacement becomes . Furthermore, in the equations that follow, as is customary, we dispense with the use of boldface symbols overdrawn with small arrows for the displacement, velocity, and acceleration vectors. We will, however, continue to convey the directions of these vectors with plus or minus signs.
Consider an object that has an initial velocity of v0 at time t0 = 0 s and moves for a time t with a constant acceleration a. For a complete description of the motion, it is also necessary to know the final velocity and displacement at time t. The final velocity v can be obtained directly from Equation 2.4:

| (2.4)

The displacement x at time t can be obtained from Equation 2.2, if a value for the average velocity can be obtained. Considering the assumption that at , we have

| (2.5)

Because the acceleration is constant, the velocity increases at a constant rate. Thus, the average velocity is midway between the initial and final velocities:

| (2.6)

Equation 2.6, like Equation 2.4, applies only if the acceleration is constant and cannot be used when the acceleration is changing. The displacement at time t can now be determined as

| (2.7)

Notice in Equations 2.4 () and 2.7 that there are five kinematic variables:

1. |
2. |
3. |
4. |
5. |

Each of the two equations contains four of these variables, so if three of them are known, the fourth variable can always be found. Example 5 illustrates how Equations 2.4 and 2.7 are used to describe the motion of an object.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 5 | | | The Displacement of a Speedboat

The speedboat in Figure 2-9 has a constant acceleration of . If the initial velocity of the boat is , find its displacement after 8.0 seconds.

Figure 2-9 (a) An accelerating speedboat. (b) The boat’s displacement x can be determined if the boat’s acceleration, initial velocity, and time of travel are known. (a, Forest Johnson/Masterfile)

Reasoning As the speedboat accelerates, its velocity is changing. The displacement of the speedboat during a given time interval is equal to the product of its average velocity during that interval and the time. The average velocity, on the other hand, is just one-half the sum of the boat’s initial and final velocities. To obtain the final velocity, we will use the definition of constant acceleration, as given in Equation 2.4.
Knowns and Unknowns The numerical values for the three known variables are listed in the table:

Description | Symbol | Value | Comment
Acceleration | a |  | Positive, because the boat is accelerating to the right, which is the positive direction. See Figure 2-9b.
Initial velocity | v0 |  | Positive, because the boat is moving to the right, which is the positive direction. See Figure 2-9b.
Time interval | t | 8.0 s |
Unknown Variable |  |  |
Displacement of boat | x | ? |

Modeling the Problem

| Step 1 Displacement
| Since the acceleration is constant, the displacement x of the boat is given by Equation 2.7, in which v0 and v are the initial and final velocities, respectively. In this equation, two of the variables, v0 and t, are known (see the Knowns and Unknowns table), but the final velocity is not. However, the final velocity can be determined by employing the definition of acceleration, as Step 2 shows.

| (2.7)

| Step 2 Acceleration
| According to Equation 2.4, which is just the definition of constant acceleration, the final velocity v of the boat is

| (2.4)

All the variables on the right side of the equals sign are known, and we can substitute this relation for v into Equation 2.7, as shown at the right.

Solution Algebraically combining the results of Steps 1 and 2, we find that

The displacement of the boat after 8.0 s is

A calculator would give the answer as 112 m, but this number must be rounded to 110 m, since the data are accurate to only two significant figures.
Related Homework: Problems 48, 68, 76

In Example 5, we combined two equations into a single equation by algebraically eliminating the final velocity v of the speedboat (which was not known). The result was the following expression for the displacement x of the speedboat:

| (2.8)

The first term (v0t) on the right side of this equation represents the displacement that would result if the acceleration were zero and the velocity remained constant at its initial value of v0. The second term () gives the additional displacement that arises because the velocity changes (a is not zero) to values that are different from its initial value. We now turn to another example of accelerated motion.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 6 | | | Catapulting a Jet

A jet is taking off from the deck of an aircraft carrier, as Figure 2-10 shows. Starting from rest, the jet is catapulted with a constant acceleration of along a straight line and reaches a velocity of . Find the displacement of the jet.

Figure 2-10 (a) A plane is being launched from an aircraft carrier. (b) During the launch, a catapult accelerates the jet down the flight deck. (a, Seth C. Peterson/Department of the Navy)

The physics of catapulting a jet from an aircraft carrier.

Reasoning When the plane is accelerating, its velocity is changing. The displacement of the plane during a given time interval is equal to the product of its average velocity during that interval and the time, the average velocity being equal to one-half the sum of the plane’s initial and final velocities. The initial and final velocities are known, but the time is not. However, the time can be determined from a knowledge of the plane’s acceleration.
Knowns and Unknowns The data for this problem are listed below:

Description | Symbol | Value | Comment
Explicit Data |  |  |
Acceleration | a |  | Positive, because the acceleration points in the positive direction. See Figure 2-10b.
Final velocity | v |  | Positive, because the final velocity points in the positive direction. See Figure 2-10b.
Implicit Data |  |  |
Initial velocity | v0 | 0 m/s | The plane starts from rest.
Unknown Variable |  |  |
Displacement of plane | x | ? |

Problem solving insight “Implicit data” are important. For instance, in Example 6 the phrase “starting from rest” means that the initial velocity is zero (v0 = 0 m/s).

Modeling the Problem

| Step 1 Displacement
| The displacement x of the jet is given by Equation 2.7, since the acceleration is constant during launch. Referring to the Knowns and Unknowns table, we see that the initial and final velocities, v0 and v, in this relation are known. The unknown time t can be determined by using the definition of acceleration (see Step 2).

| (2.7)

| Step 2 Acceleration
| The acceleration of an object is defined by Equation 2.4 as the change in its velocity, , divided by the elapsed time t:

| (2.4)

Solving for t yields

All the variables on the right side of the equals sign are known, and we can substitute this result for t into Equation 2.7, as shown at the right.

Solution Algebraically combining the results of Steps 1 and 2, we find that

The displacement of the jet is

Related Homework: Problems 23, 44, 50

In Example 6 we were able to determine the displacement x of the jet from a knowledge of its acceleration a and its initial and final velocities, v0 and v. The result was . Solving for v2 gives

| (2.9)

Equation 2.9 is often used when the time involved in the motion is unknown.
Table 2-1 presents a summary of the equations that we have been considering. These equations are called the equations of kinematics. Each equation contains four of the five kinematic variables, as indicated by the check marks (✓) in the table. The next section shows how to apply the equations of kinematics.

Table 2-1 Equations of Kinematics for Constant Acceleration

|  | Variable
Equation Number | Equation | x | a | v | v0 | t
(2.4) |  | – | ✓ | ✓ | ✓ | ✓
(2.7) |  | ✓ | – | ✓ | ✓ | ✓
(2.8) |  | ✓ | ✓ | – | ✓ | ✓
(2.9) |  | ✓ | ✓ | ✓ | ✓ | –

Check Your Understanding 3

A motorcycle starts from rest and has a constant acceleration. In a certain time interval, its displacement triples. In the same time interval, by what factor does its velocity increase? (The answer is given at the end of the book.) Background: When the acceleration is constant, the equations of kinematics given in Table 2-1 apply.
For similar questions (including calculational counterparts), consult Self-Assessment Test 2.2. This test is described at the end of Section 2.5.

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