28.5

Relativistic Momentum

Thus far we have discussed how time intervals and distances between two events are measured by observers moving at a constant velocity relative to each other. Special relativity also alters our ideas about momentum and energy.
Recall from Section 7.2 that when two or more objects interact, the principle of conservation of linear momentum applies if the system of objects is isolated. This principle states that the total linear momentum of an isolated system remains constant at all times. (An isolated system is one in which the sum of the external forces acting on the objects is zero.) The conservation of linear momentum is a law of physics and, in accord with the relativity postulate, is valid in all inertial reference frames. That is, when the total linear momentum is conserved in one inertial reference frame, it is conserved in all inertial reference frames.
As an example of momentum conservation, suppose several people are watching two billiard balls collide on a frictionless pool table. One person is standing next to the pool table, and the other is moving past the table with a constant velocity. Since the two balls constitute an isolated system, the relativity postulate requires that both observers must find the total linear momentum of the two-ball system to be the same before, during, and after the collision. For this kind of situation, Section 7.1 defines the classical linear momentum of an object to be the product of its mass m and velocity . As a result, the magnitude of the classical momentum is p = mv. As long as the speed of an object is considerably smaller than the speed of light, this definition is adequate. However, when the speed approaches the speed of light, an analysis of the collision shows that the total linear momentum is not conserved in all inertial reference frames if one defines linear momentum simply as the product of mass and velocity. In order to preserve the conservation of linear momentum, it is necessary to modify this definition. The theory of special relativity reveals that the magnitude of the relativistic momentum must be defined as in Equation 28.3:

| (28.3)

The total relativistic momentum of an isolated system is conserved in all inertial reference frames.
From Equation 28.3, we can see that the magnitudes of the relativistic and nonrela-tivistic momenta differ by the same factor of that occurs in the time-dilation and length-contraction equations. Since this factor is always less than 1 and occurs in the denominator in Equation 28.3, the relativistic momentum is always larger than the non-relativistic momentum. To illustrate how the two quantities differ as the speed v increases, Figure 28-7 shows a plot of the ratio of the momentum magnitudes (relativistic/nonrelativistic) as a function of v. According to Equation 28.3, this ratio is just . The graph shows that for speeds attained by ordinary objects, such as cars and planes, the relativistic and nonrelativistic momenta are almost equal because their ratio is nearly 1. Thus, at speeds much less than the speed of light, either the nonrelativistic momentum or the relativistic momentum can be used to describe collisions. On the other hand, when the speed of the object becomes comparable to the speed of light, the relativistic momentum becomes significantly greater than the nonrelativistic momentum and must be used. Example 5 deals with the relativistic momentum of an electron traveling close to the speed of light.

Figure 28-7 This graph shows how the ratio of the magnitude of the relativistic momentum to the magnitude of the nonrelativistic momentum increases as the speed of an object approaches the speed of light.

| Example 5 | | | The Relativistic Momentum of a High-Speed Electron

The particle accelerator at Stanford University (Figure 28-8) is 3 km long and accelerates electrons to a speed of 0.999 999 999 7c, which is very nearly equal to the speed of light. Find the magnitude of the relativistic momentum of an electron that emerges from the accelerator, and compare it with the nonrelativistic value.

Figure 28-8 The Stanford 3-km linear accelerator accelerates electrons almost to the speed of light. (Bill Marsh/Photo Researchers, Inc.)

Reasoning and Solution The magnitude of the electron’s relativistic momentum can be obtained from Equation 28.3 if we recall that the mass of an electron is :

This value for the magnitude of the momentum agrees with that measured experimentally. The relativistic momentum is greater than the nonrelativistic momentum by a factor of

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