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Relativistic kinetic energy

Relativistic kinetic energy of any particle of mass m is

K rel = ( γ 1 ) m c 2 .

When an object is motionless, its speed is u = 0 and

γ = 1 1 u 2 c 2 = 1

so that K rel = 0 at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical 1 2 m u 2 . To show that the expression for K rel reduces to the classical expression for kinetic energy at low speeds, we use the binomial expansion to obtain an approximation for ( 1 + ε ) n valid for small ε :

( 1 + ε ) n = 1 + n ε + n ( n 1 ) 2 ! ε 2 + n ( n 1 ) ( n 2 ) 3 ! ε 3 + 1 + n ε

by neglecting the very small terms in ε 2 and higher powers of ε . Choosing ε = u 2 / c 2 and n = 1 2 leads to the conclusion that γ at nonrelativistic speeds, where ε = u / c is small, satisfies

γ = ( 1 u 2 / c 2 ) −1 / 2 1 + 1 2 ( u 2 c 2 ) .

A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small speed here, most terms are very small. Thus, the expression derived here for γ is not exact, but it is a very accurate approximation. Therefore, at low speed:

γ 1 = 1 2 ( u 2 c 2 ) .

Entering this into the expression for relativistic kinetic energy gives

K rel = [ 1 2 ( u 2 c 2 ) ] m c 2 = 1 2 m u 2 = K class .

That is, relativistic kinetic energy becomes the same as classical kinetic energy when u < < c .

It is even more interesting to investigate what happens to kinetic energy when the speed of an object approaches the speed of light. We know that γ becomes infinite as u approaches c , so that K rel also becomes infinite as the velocity approaches the speed of light ( [link] ). The increase in K rel is far larger than in K class as v approaches c. An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.

The speed of light

No object with mass can attain the speed of light    .

The speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than c always add to less than c . Both the relativistic form for kinetic energy and the ultimate speed limit being c have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach—not reach—the speed of light.

This is a graph of the kinetic energy as a function of speed. Two curves are shown: the relativistic kinetic energy and the classical kinetic energy. Both curves are small at low speeds. The relativistic energy rises faster than the classical energy and has a vertical asymptote at u=c. The classical energy crosses u=c at a finite value and continues to increase but remains finite for u>c.
This graph of K rel versus velocity shows how kinetic energy increases without bound as velocity approaches the speed of light. Also shown is K class , the classical kinetic energy.

Comparing kinetic energy

An electron has a velocity v = 0.990 c . (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is 9.11 × 10 −31 kg . )

Strategy

The expression for relativistic kinetic energy is always correct, but for (a), it must be used because the velocity is highly relativistic (close to c ). First, we calculate the relativistic factor γ , and then use it to determine the relativistic kinetic energy. For (b), we calculate the classical kinetic energy (which would be close to the relativistic value if v were less than a few percent of c ) and see that it is not the same.

Questions & Answers

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
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
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
ganesh
you are correct
Priyojit
law of multiple
Wahid
if we heated the ice then the refractive index be change from natural water
Nepal
can someone explain normalization condition
Priyojit Reply
please tell
Swati
yes
Chemist
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
Gitanjali
The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
What is photoelectric effect
Amit Reply
it gives practical evidence of particke nature of light.
Omsai
particle nature
Omsai
photoelectric effect is the phenomenon of emission of electrons from a material(i.e Metal) when it is exposed to sunlight. Emitted electrons are called as photo electrons.
Anil
what are the applications of quantum mechanics to medicine?
Neptune
application of quantum mechanics in medicine: 1) improved disease screening and treatment ; using a relatively new method known as BIO- BARCODE ASSAY we can detect disease-specific clues in our blood using gold nanoparticles. 2) in Genomic medicine 3) in protein folding 4) in radio theraphy(MRI)
Anil
Quantam physics ki basic concepts?
Laxmikanta Reply
why does not electron exits in nucleaus
Kabbo Reply
electrons have negative
YASH
Proton and meltdown has greater mass than electron. So it naturally electron will move around nucleus such as gases surrounded earth
Amalesh
.......proton and neutron....
Amalesh
excuse me yash what negative
Rika
coz, electron contained minus ion
Manish
negative sign rika shrestha ji
YASH
electron is the smallest negetive charge...An anaion i.e., negetive ion contains extra electrons. How ever an atom is neutral so it must contains proton and electron
Amalesh
yes yash ji
Rika
yes friends
Prema
koantam theory
Laxmikanta
yes prema
Rika
quantum theory tells us that both light and matter consists of tiny particles which have wave like propertise associated with them.
Prema
proton and nutron nuclear power is best than proton and electron kulamb force
Laxmikanta
what is de-broglie wave length?
Ramsuphal
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
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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