# Relativistic Momentum, Energy

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Relativistic MomentumSummary:

Calculate relativistic momentum.

Explain why the only mass it makes sense to talk about is rest mass.

Figure 1: Momentum is an important concept forthese football players from the University ofCalifornia at Berkeley and the University ofCalifornia at Davis. Players with more mass oftenhave a larger impact because their momentum islarger. For objects moving at relativistic speeds,the effect is even greater. (credit: John MartinezPavliga)

In classical physics, momentum is a simple product of mass and velocity. However, we saw in thelast 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 Newtons 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

ofWork, Energy, and Energy Resourcesis 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.

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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 momentumpis classical momentum multiplied by the relativistic factor .

p=mu,

(1)

where mis the rest massof the object, uis its velocity relative to an observer, and the

relativistic factor

=11u2c2.

(2)

Note that we use ufor velocity here to distinguish it from relative velocity vbetween observers.

Only one observer is being considered here. Withpdefined in this way, total momentumptotis

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 mubecomes the classical muat low velocities, because 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 , relativistic momentum approaches

infinity as uapproaches c. (SeeFigure 2.) This is another indication that an object with mass cannotreach the speed of light. If it did, its momentum would become infinite, an unreasonable value.

Figure 2: Relativistic momentum approaches

infinity as the velocity of an object approaches

the speed of light.

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MISCONCEPTION ALERT: RELATIVISTIC MASS AND MOMENTUM:

The relativistically correct definition of momentum asp=muis sometimes taken to imply that

mass varies with velocity: mvar=m, particularly in older textbooks. However, note that mis themass of the object as measured by a person at rest relative to the object. Thus, mis defined to

be the rest mass, which could be measured at rest, perhaps using gravity. When a mass ismoving 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.

InRelativistic 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.

What is the momentum of an electron traveling at a speed 0.985c? The rest mass of the electron

is 9.111031kg.

[ Show ]

Section Summary The law of conservation of momentum is valid whenever the net external force is zero and

for relativistic momentum. Relativistic momentumpis classical momentum multiplied by the

relativistic factor .

p=mu, where mis the rest mass of the object, uis its velocity relative to an observer, and

the relativistic factor =11u2c2.

At low velocities, relativistic momentum is equivalent to classical momentum.

Relativistic momentum approaches infinity as uapproaches c. This implies that an object

with mass cannot reach the speed of light. Relativistic momentum is conserved, just as classical momentum is conserved.

Conceptual QuestionsExercise 1

How does modern relativity modify the law of conservation of momentum?

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Exercise 2

Is it possible for an external force to be acting on a system and relativistic momentum to be

conserved? Explain.

Problem ExercisesExercise 1Find the momentum of a helium nucleus having a mass of 6.681027kgthat is moving at 0.200c.

Solut ion

4.091019kgm/s

Exercise 2

What is the momentum of an electron traveling at 0.980c?

Exercise 3

(a) Find the momentum of a 1.00109kgasteroid heading towards the Earth at 30.0 km/s. (b)

Find the ratio of this momentum to the classical momentum. (Hint: Use the approximation

that =1+(1/2)v2/c2at low velocities.)

Solut ion

(a) 3.0000000151013kgm/s.

(b) Ratio of relativistic to classical momenta equals 1.000000005 (extra digits to show small effects)

Exercise 4

(a) What is the momentum of a 2000 kg satellite orbiting at 4.00 km/s? (b) Find the ratio of this

momentum to the classical momentum. (Hint: Use the approximation that =1+(1/2)v2/c2atlow velocities.)Exercise 5

What is the velocity of an electron that has a momentum of 3.041021kgm/s? Note that you must

calculate the velocity to at least four digits to see the difference from c.

Solut ion

2.9957108m/s

Exercise 6

Find the velocity of a proton that has a momentum of 4.481019kgm/s.

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Exercise 7

(a) Calculate the speed of a 1.00-gparticle of dust that has the same momentum as a proton

moving at 0.999c. (b) What does the small speed tell us about the mass of a proton compared to

even a tiny amount of macroscopic matter?

Solut ion

(a) 1.121108m/s

(b) The small speed tells us that the mass of a proton is substantially smaller than that of even a tiny

amount of macroscopic matter!

Solut ion

(a) 1.121108m/s

(b) The small speed tells us that the mass of a proton is substantially smaller than that of even a tiny

amount of macroscopic matter!

Exercise 8

(a) Calculate for a proton that has a momentum of 1.00 kgm/s.(b) What is its speed? Such

protons form a rare component of cosmic radiation with uncertain origins.

Glossary

relativistic momentum:

p, the momentum of an object moving at relativistic velocity;p=mu, where mis the rest

mass of the object, uis its velocity relative to an observer, and the relativistic

factor =11u2c2

rest mass:

the mass of an object as measured by a person at rest relative to the object

Relativistic Energy

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