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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 . )


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

in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
Kimani Reply
derivation of lateral shieft
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total binding energy of ionic crystal at equilibrium is
All Reply
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Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
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Izevbogie Reply
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Radioactive substance
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
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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.
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
this msg is out of mistake. sorry friends​.
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
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?
physics is the study of matter,energy and their interactions
all living things are matter
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explain l-s coupling
Depk Reply
Practice Key Terms 4

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