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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.

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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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