28.5 Relativistic momentum

 Page 1 / 3
• Calculate relativistic momentum.
• Explain why the only mass it makes sense to talk about is rest mass.

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 $p$ is classical momentum multiplied by the relativistic factor $\gamma$ .

$p=\text{γmu},$

where $m$ is the rest mass    of the object, $u$ is its velocity relative to an observer, and the relativistic factor

$\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}.$

Note that we use $u$ for velocity here to distinguish it from relative velocity $v$ between observers. Only one observer is being considered here. With $p$ defined in this way, total momentum ${p}_{\text{tot}}$ 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 $\text{γmu}$ becomes the classical $\text{mu}$ at low velocities, because $\gamma$ 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 $\gamma$ , relativistic momentum approaches infinity as $u$ approaches $c$ . (See [link] .) 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.

Misconception alert: relativistic mass and momentum

The relativistically correct definition of momentum as $p=\text{γmu}$ is sometimes taken to imply that mass varies with velocity: ${m}_{\text{var}}=\mathrm{\gamma m}$ , particularly in older textbooks. However, note that $m$ is the mass of the object as measured by a person at rest relative to the object. Thus, $m$ 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.

What is the momentum of an electron traveling at a speed $0\text{.}\text{985}c$ ? The rest mass of the electron is $9\text{.}\text{11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .

$p=\gamma \text{mu}=\frac{\text{mu}}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}=\frac{\left(9\text{.}\text{11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(0\text{.}\text{985}\right)\left(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)}{\sqrt{1-\frac{\left(0.985c{\right)}^{2}}{{c}^{2}}}}=1.56×{\text{10}}^{-\text{21}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$

Section summary

• The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum. Relativistic momentum $p$ is classical momentum multiplied by the relativistic factor $\gamma$ .
• $p=\text{γmu}$ , where $m$ is the rest mass of the object, $u$ is its velocity relative to an observer, and the relativistic factor $\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}$ .
• At low velocities, relativistic momentum is equivalent to classical momentum.
• Relativistic momentum approaches infinity as $u$ approaches $c$ . This implies that an object with mass cannot reach the speed of light.
• Relativistic momentum is conserved, just as classical momentum is conserved.

Conceptual questions

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

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

Problem exercises

Find the momentum of a helium nucleus having a mass of $6\text{.}\text{68}×{\text{10}}^{\text{–27}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ that is moving at $0\text{.}\text{200}c$ .

$4\text{.}\text{09}×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$

What is the momentum of an electron traveling at $0\text{.}\text{980}c$ ?

(a) Find the momentum of a $1\text{.}\text{00}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{kg}$ asteroid heading towards the Earth at $\text{30.0 km/s}$ . (b) Find the ratio of this momentum to the classical momentum. (Hint: Use the approximation that $\gamma =1+\left(1/2\right){v}^{2}/{c}^{2}$ at low velocities.)

(a) $3\text{.}\text{000000015}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}$ .

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

(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 $\gamma =1+\left(1/2\right){v}^{2}/{c}^{2}$ at low velocities.)

What is the velocity of an electron that has a momentum of $3.04×{\text{10}}^{\text{–21}}\phantom{\rule{0.25em}{0ex}}\text{kg⋅m/s}$ ? Note that you must calculate the velocity to at least four digits to see the difference from $c$ .

$\text{2.9957}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

Find the velocity of a proton that has a momentum of $4\text{.}\text{48}×{\text{–10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{kg⋅m/s}\text{.}$

(a) Calculate the speed of a $1.00-\mu \text{g}$ particle of dust that has the same momentum as a proton moving at $0\text{.}\text{999}c$ . (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

(a) $1\text{.}\text{121}×{\text{10}}^{\text{–8}}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

(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!

(a) Calculate $\gamma$ for a proton that has a momentum of $\text{1.00 kg⋅m/s}\text{.}$ (b) What is its speed? Such protons form a rare component of cosmic radiation with uncertain origins.

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
What is thermal heat all about
why uniform circular motion is called a periodic motion?.
when a train start from A & it returns at same station A . what is its acceleration?
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
what are the types of radioactivity
Worku
what is static friction
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
definition of mass of conversion
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
Boniface
the range of objects and phenomena studied in physics is
Boniface