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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

How does, ray of light coming form focus, behaves in concave mirror after refraction?
Bishesh Reply
Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
Sushant
What is motion
Izevbogie Reply
Anything which changes itself with respect to time or surrounding
Sushant
good
Chemist
and what's time? is time everywhere same
Chemist
No
Sushant
how can u say that
Chemist
do u know about black hole
Chemist
Not so more
Sushant
Radioactive substance
DHEERAJ
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
Sushant
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
DHEERAJ
explain cavendish experiment to determine the value of gravitational concept.
Celine Reply
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
Alwielland
acceleration also inc
Usman
which paper will be subjective and which one objective
jay
normal distributiin of errors report
Dennis
normal distribution of errors
Dennis
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
Rafi
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
tahreem
physics is the study of matter,energy and their interactions
Buvanes
all living things are matter
Buvanes
why rolling friction is less than sliding friction
tahreem
thanks buvanas
tahreem
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat 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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