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Relativistic energy and momentum

We know classically that kinetic energy and momentum are related to each other, because:

K class = p 2 2 m = ( m u ) 2 2 m = 1 2 m u 2 .

Relativistically, we can obtain a relationship between energy and momentum by algebraically manipulating their defining equations. This yields:

E 2 = ( p c ) 2 + ( m c 2 ) 2 ,

where E is the relativistic total energy, E = m c 2 / 1 u 2 / c 2 , and p is the relativistic momentum. This relationship between relativistic energy and relativistic momentum is more complicated than the classical version, but we can gain some interesting new insights by examining it. First, total energy is related to momentum and rest mass. At rest, momentum is zero, and the equation gives the total energy to be the rest energy m c 2 (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum p increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term ( m c 2 ) 2 becomes negligible compared with the momentum term ( p c ) 2 ; thus, E = p c at extremely relativistic velocities.

If we consider momentum p to be distinct from mass, we can determine the implications of the equation E 2 = ( p c ) 2 + ( m c 2 ) 2 , for a particle that has no mass. If we take m to be zero in this equation, then E = p c , or p = E / c . Massless particles have this momentum. There are several massless particles found in nature, including photons (which are packets of electromagnetic radiation). Another implication is that a massless particle must travel at speed c and only at speed c . It is beyond the scope of this text to examine the relationship in the equation E 2 = ( p c ) 2 + ( m c 2 ) 2 in detail, but you can see that the relationship has important implications in special relativity.

Check Your Understanding What is the kinetic energy of an electron if its speed is 0.992 c ?

K rel = ( γ 1 ) m c 2 = ( 1 1 u 2 c 2 1 ) m c 2 = ( 1 1 ( 0.992 c ) 2 c 2 1 ) ( 9.11 × 10 31 kg ) ( 3.00 × 10 8 m/s ) 2 = 5.67 × 10 13 J

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Summary

  • The relativistic work-energy theorem is W net = E E 0 = γ m c 2 m c 2 = ( γ 1 ) m c 2 .
  • Relativistically, W net = K rel where K rel is the relativistic kinetic energy.
  • An object of mass m at velocity u has kinetic energy K rel = ( γ 1 ) m c 2 , where γ = 1 1 u 2 c 2 .
  • At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
  • No object with mass can attain the speed of light, because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
  • Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
  • The total energy of a particle with mass m traveling at speed u is defined as E = γ m c 2 , where γ = 1 1 u 2 c 2 and u denotes the velocity of the particle.
  • The rest energy of an object of mass m is E 0 = m c 2 , meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
  • We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy. The equation E 2 = ( p c ) 2 + ( m c 2 ) 2 relates the relativistic total energy E and the relativistic momentum p . At extremely high velocities, the rest energy m c 2 becomes negligible, and E = p c .

Questions & Answers

plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
Ime Reply
expression for photon as wave
BARISUA Reply
Are beta particle and eletron are same?
Amalesh Reply
yes
mari
how can you confirm?
Amalesh
sry
Saiaung
If they are same then why they named differently?
Amalesh
because beta particles give the information that the electron is ejected from the nucleus with very high energy
Absar
what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
what is CHEMISTRY
trpathy Reply
it's a subject
Akhter
it's a branch in science which deals with the properties,uses and composition of matter
Eniabire
what is a Higgs Boson please?
FRANKLINE Reply
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
M.D
bro little abit getting confuse if i am wrong than please clarify me
M.D
the law of refraction of direct current lines at the boundary between two conducting media of
BATTULA Reply
what is the black body of an ideal radiator
Areej Reply
uncertainty principles is applicable to
Areej
fermions
FRANKLINE
what is the cause of the expanding universe?
FRANKLINE
microscopic particles or gases
Areej
Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space.
Areej
Thanks for your contribution Areej.
FRANKLINE
no problem
Areej
what is photoelectric equation
HIMANSHU Reply
How does fringe intensity depend upon slit width in single slit diffraction?
Abhishek Reply
intensity seems to be directly proportional radius of slit
Mathieu
what are the applications of Bernoulli's equation
Shaukat
VOLTE
Md Reply
what is Draic equation
M.D
about nuclear angular momentum
rahul Reply
what is spin
MUKESH Reply
Practice Key Terms 4

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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