<< Chapter < Page Chapter >> Page >
  • Calculate relativistic momentum.
  • Explain why the only mass it makes sense to talk about is rest mass.
Action photo from a college football game.
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. (credit: 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 p size 12{p} {} is classical momentum multiplied by the relativistic factor γ size 12{γ} {} .

p = γmu , size 12{p= ital "γmu"} {}

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

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

Note that we use u size 12{u} {} for velocity here to distinguish it from relative velocity v size 12{v} {} between observers. Only one observer is being considered here. With p size 12{p} {} defined in this way, total momentum p tot 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 γmu size 12{ ital "γmu"} {} becomes the classical mu size 12{ ital "mu"} {} at low velocities, because γ 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 γ size 12{γ} {} , relativistic momentum approaches infinity as u size 12{u} {} approaches c size 12{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.

 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 moving upward along the vertical line at x is equal to one point zero c. This graph shows that relativistic momentum approaches infinity as the velocity of an object approaches the speed of light.
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 = γmu size 12{p= ital "γmu"} {} is sometimes taken to imply that mass varies with velocity: m var = γm size 12{m rSub { size 8{"var"} } =γm} {} , particularly in older textbooks. However, note that m size 12{m} {} is the mass of the object as measured by a person at rest relative to the object. Thus, m 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.

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

Answer

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
Got questions? Get instant answers now!

Section summary

  • The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum. Relativistic momentum p size 12{p} {} is classical momentum multiplied by the relativistic factor γ size 12{γ} {} .
  • p = γmu size 12{p= ital "γmu"} {} , where m size 12{m} {} is the rest mass of the object, u size 12{u} {} is its velocity relative to an observer, and the relativistic factor γ = 1 1 u 2 c 2 size 12{γ= { {1} over { sqrt {1 - { {u rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {} .
  • At low velocities, relativistic momentum is equivalent to classical momentum.
  • Relativistic momentum approaches infinity as u size 12{u} {} approaches c size 12{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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Problem exercises

Find the momentum of a helium nucleus having a mass of 6 . 68 × 10 –27 kg size 12{6 "." "68" times "10" rSup { size 8{"27"} } " kg"} {} that is moving at 0 . 200 c size 12{0 "." "200"c} {} .

4 . 09 × 10 –19 kg m/s

Got questions? Get instant answers now!

What is the momentum of an electron traveling at 0 . 980 c size 12{0 "." "980"c} {} ?

Got questions? Get instant answers now!

(a) Find the momentum of a 1 . 00 × 10 9 kg size 12{1 "." "00" times "10" rSup { size 8{9} } " kg"} {} asteroid heading towards the Earth at 30.0 km/s size 12{"30" "." 0" km/s"} {} . (b) Find the ratio of this momentum to the classical momentum. (Hint: Use the approximation that γ = 1 + ( 1 / 2 ) v 2 / c 2 size 12{γ=1+ \( 1/2 \) v rSup { size 8{2} } /c rSup { size 8{2} } } {} at low velocities.)

(a) 3 . 000000015 × 10 13 kg m/s size 12{ {underline {3 "." "000000015 " times "10" rSup { size 8{"13"} } " kg" cdot "m/s"}} } {} .

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

Got questions? Get instant answers now!

(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 γ = 1 + ( 1 / 2 ) v 2 / c 2 size 12{γ=1+ \( 1/2 \) v rSup { size 8{2} } /c rSup { size 8{2} } } {} at low velocities.)

Got questions? Get instant answers now!

What is the velocity of an electron that has a momentum of 3.04 × 10 –21 kg⋅m/s size 12{3 "." "34" times "10" rSup { size 8{"21"} } " kg" "." "m/s?"} {} ? Note that you must calculate the velocity to at least four digits to see the difference from c .

2.9957 × 10 8 m/s size 12{ {underline {2 "." "988" times "10" rSup { size 8{8} } " m/s"}} } {}

Got questions? Get instant answers now!

Find the velocity of a proton that has a momentum of 4 . 48 × –10 - 19 kg⋅m/s . size 12{4 "." "48" times "10" rSup { size 8{"19"} } " kg" "." "m/s" "." } {}

Got questions? Get instant answers now!

(a) Calculate the speed of a 1.00- μ g particle of dust that has the same momentum as a proton moving at 0 . 999 c size 12{0 "." "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 . 121 × 10 –8 m/s size 12{1 "." "121" times "10" rSup { size 8{"-8"} } " 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!

Got questions? Get instant answers now!

(a) Calculate γ size 12{γ} {} for a proton that has a momentum of 1.00 kg⋅m/s . size 12{1 "." "00 kg" "." "m/s" "." } {} (b) What is its speed? Such protons form a rare component of cosmic radiation with uncertain origins.

Got questions? Get instant answers now!

Questions & Answers

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
Atega Reply
introduction to econometrics
Husseini Reply
Hi
mostafa
hi
LEMLEM
hello
Sammol
hi
Mahesh
bi
Ruqayat
hi
Ruqayat
Hi fellas
Nyawa
hey
Sammol
hi
God
hello
Jahara
Good morning
Jorge
hi
abubakar
hi
Nmesoma
hi
Mahesh
Hi
Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
explain inflationcause and itis degre
Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
Mercy Reply
Hi
Isiaka
Hi
Geli
hy
Man
👋
Bahunda
hy how are you?
Man
ys
Amisha
how are you guys
Sekou
f9 guys
Amisha
how are you guys
Sekou
ys am also fine
Amisha
fine and you guys
Geli
from Nepal
Amisha
nawalparasi district from belatari
Amisha
nd u
Amisha
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask