Learning Objectives

Learning Objectives

By the end of this section, you will be able to do the following:

  • Calculate relativistic momentum
  • Explain why the only mass it makes sense to talk about is rest mass
Action photo from a college football game.
Figure 11.18 Momentum is an important concept for these football players from the University of California at Berkeley and the University of California at Davis. Players with more mass often have a larger impact because their momentum is larger. For objects moving at relativistic speeds, the effect is even greater. (John Martinez Pavliga)

In classical physics, momentum is a simple product of mass and velocity. However, we saw in the last section that when special relativity is taken into account, massive objects have a speed limit. What effect do you think mass and velocity have on the momentum of objects moving at relativistic speeds?

Momentum is one of the most important concepts in physics. The broadest form of Newton’s second law is stated in terms of momentum. Momentum is conserved whenever the net external force on a system is zero. This makes momentum conservation a fundamental tool for analyzing collisions. All of Work, Energy, and Energy Resources is devoted to momentum, and momentum has been important for many other topics as well, particularly where collisions were involved. We will see that momentum has the same importance in modern physics. Relativistic momentum is conserved, and much of what we know about subatomic structure comes from the analysis of collisions of accelerator-produced relativistic particles.

The first postulate of relativity states that the laws of physics are the same in all inertial frames. Does the law of conservation of momentum survive this requirement at high velocities? The answer is yes, provided that the momentum is defined as follows.

Relativistic Momentum

Relativistic momentum pp size 12{p} {} is classical momentum multiplied by the relativistic factor γγ size 12{γ} {}

11.48 p=γmu,p=γmu, size 12{p= ital "γmu"} {}

where mm size 12{m} {} is the rest mass of the object, uu size 12{u} {} is its velocity relative to an observer, and the relativistic factor

11.49 γ=11u2c2.γ=11u2c2 size 12{γ= { {1} over { sqrt {1 - { {u rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}.

Note that we use uu size 12{u} {} for velocity here to distinguish it from relative velocity vv size 12{v} {} between observers. Only one observer is being considered here. With pp size 12{p} {} defined in this way, total momentum ptotptot size 12{p rSub { size 8{"tot"} } } {} is conserved whenever the net external force is zero, just as in classical physics. Again we see that the relativistic quantity becomes virtually the same as the classical at low velocities. That is, relativistic momentum γmuγmu size 12{ ital "γmu"} {} becomes the classical mumu size 12{ ital "mu"} {} at low velocities, because γγ size 12{γ} {} is very nearly equal to 1 at low velocities.

Relativistic momentum has the same intuitive feel as classical momentum. It is greatest for large masses moving at high velocities, but, because of the factor γγ size 12{γ} {}, relativistic momentum approaches infinity as uu size 12{u} {} approaches c.c. size 12{c} {} (See Figure 11.19.) This is another indication that an object with mass cannot reach the speed of light. If it did, its momentum would become infinite, an unreasonable value.

In this figure a graph is shown on a coordinate system of axes. The x-axis is labelled as speed u meter per second. On x-axis velocity of the object is shown in terms of the speed of light starting from zero at origin to one point zero c where c is the speed of light. The y-axis is labelled as momentum p rel kilogram meter per second. On y-axis relativistic momentum is shown in terms of kilogram meter per starting from zero at origin to four point zero. The graph in the given figure is concave up and movi
Figure 11.19 Relativistic momentum approaches infinity as the velocity of an object approaches the speed of light.

Misconception Alert: Relativistic Mass and Momentum

The relativistically correct definition of momentum as p=γmup=γmu size 12{p= ital "γmu"} {} is sometimes taken to imply that mass varies with velocity: mvar=γmmvar=γm size 12{m rSub { size 8{"var"} } =γm} {}, particularly in older textbooks. However, note that mm size 12{m} {} is the mass of the object as measured by a person at rest relative to the object. Thus, mm size 12{m} {} is defined to be the rest mass, which could be measured at rest, perhaps using gravity. When a mass is moving relative to an observer, the only way that its mass can be determined is through collisions or other means in which momentum is involved. Since the mass of a moving object cannot be determined independently of momentum, the only meaningful mass is rest mass. Thus, when we use the term mass, assume it to be identical to rest mass.

Relativistic momentum is defined in such a way that the conservation of momentum will hold in all inertial frames. Whenever the net external force on a system is zero, relativistic momentum is conserved, just as is the case for classical momentum. This has been verified in numerous experiments.

In Relativistic Energy, the relationship of relativistic momentum to energy is explored. That subject will produce our first inkling that objects without mass may also have momentum.

Check Your Understanding
Exercise 3

What is the momentum of an electron traveling at a speed 0.985c?0.985c? size 12{0 "." "985"c} {} The rest mass of the electron is 9.11×1031 kg.9.11×1031 kg. size 12{9 "." "11" times "10" rSup { size 8{ - "31"} } " kg"} {}

11.50 p = γ mu = mu 1 u 2 c 2 = ( 9 . 11 × 10 31 kg ) ( 0 . 985 ) ( 3.00 × 10 8 m/s ) 1 ( 0.985 c ) 2 c 2 = 1.56 × 10 21 kg m/s p = γ mu = mu 1 u 2 c 2 = ( 9 . 11 × 10 31 kg ) ( 0 . 985 ) ( 3.00 × 10 8 m/s ) 1 ( 0.985 c ) 2 c 2 = 1.56 × 10 21 kg m/s