7.6

Concepts & Calculations

Momentum and energy are two of the most fundamental concepts in physics. As we have seen in this chapter, momentum is a vector and, like all vectors, has a magnitude and a direction. In contrast, energy is a scalar quantity, as Chapter 6 discusses, and does not have a direction associated with it. Concepts & Calculations Example 10 provides the opportunity to review how the vector nature of momentum and the scalar nature of kinetic energy influence calculations using these quantities.

| Concepts & Calculations Example 10 | | | A Scalar and a Vector

Two joggers, Jim and Tom, are both running at a speed of 4.00 m/s. Jim has a mass of 90.0 kg, and Tom has a mass of 55.0 kg. Find the kinetic energy and momentum of the two-jogger system when (a) Jim and Tom are both running due north (Figure 7-19a) and (b) Jim is running due north and Tom is running due south (Figure 7-19b).

Figure 7-19 (a) Two joggers are running due north with the same velocities. (b) One jogger is running due north and the other due south, with velocities that have the same magnitudes but different directions.

Concept Questions and Answers Does the total kinetic energy have a smaller value in case (a) or (b), or is it the same in both cases?

Answer Everything is the same in both cases, except that both joggers are running due north in (a), but one is running due north and one is running due south in (b). Kinetic energy is a scalar quantity and does not depend on directional differences such as these. Therefore, the kinetic energy of the two-jogger system is the same in both cases.

Does the total momentum have a smaller magnitude in case (a) or (b), or is it the same for each?

Answer Momentum is a vector. As a result, when we add Jim’s momentum to Tom’s momentum to get the total for the two-jogger system, we must take the directions into account. For example, we can use positive to denote north and negative to denote south. In case (a) two positive momentum vectors are added, while in case (b) one positive and one negative vector are added. Due to the partial cancellation in case (b), the resulting total momentum will have a smaller magnitude than in case (a).

Solution

a. | The kinetic energy of the two-jogger system can be obtained by applying the definition of kinetic energy as mv2 (see Equation 6.2) to each jogger:

The momentum of each jogger is his mass times his velocity, according to the definition given in Equation 7.2. Using Psystem to denote the momentum of the two-jogger system and denoting due north as the positive direction, we have

b. | In these calculations, we note that Tom’s velocity is in the due south direction, so it is − 4.00 m/s, not + 4.00 m/s. Proceeding as in part (a), we find

Momentum and kinetic energy are not the same concepts, and the next example explores further some of the differences between them.

| Concepts & Calculations Example 11 | | | Momentum and Kinetic Energy

The following table gives mass and speed data for the two objects in Figure 7-20. Find the magnitude of the momentum and the kinetic energy for each object.

| Mass | Speed
Object A | 2.0 kg | 6.0 m/s
Object B | 6.0 kg | 2.0 m/s

Figure 7-20 The two objects have different masses and speeds. Are their momenta and kinetic energies the same?

Concept Questions and Answers Is it possible for two objects to have different speeds when their momenta have the same magnitude?

Answer Yes. The magnitude of an object’s momentum is the product of its mass and its speed, according to Equation 7.2. Both mass and speed play a role. The speed can be reduced, for instance, but if the mass is increased proportionally, the magnitude of the momentum will be the same.

If two objects have the same momentum, do they necessarily have the same kinetic energy?

Answer No. Kinetic energy is Momentum is , which means that . Substituting this relation into the kinetic energy expression shows that . Thus, two objects can have the same momentum p, but if each has different mass m, their kinetic energies will be different.

Solution Using Equation 7.2 for the momentum and Equation 6.2 for the kinetic energy, we find that

Need more practice?

Interactive LearningWare7.3
A 1080-kg car moves with a speed of 28.0 m/s and is headed 30.0° north of east. A second car has a mass of 1630 kg and is headed due south. A third car has a mass of 1350 kg and is headed due west. What are the speeds of the second and third cars if the total linear momentum of the three-car system is zero?
At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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