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Kinetic energy and the ultimate speed limit

Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression 1 2 mv 2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {} . The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is

W net = KE . size 12{W rSub { size 8{"net"} } ="KE"} {}

Relativistically, at rest we have rest energy E 0 = mc 2 . The work increases this to the total energy E = γmc 2 . Thus,

W net = E E 0 = γ mc 2 mc 2 = γ 1 mc 2 .

Relativistically, we have W net = KE rel size 12{W="KE" rSub { size 8{"rel"} } } {} .

Relativistic kinetic energy

Relativistic kinetic energy is

KE rel = γ 1 mc 2 . size 12{"KE" rSub { size 8{"rel"} } = left (γ - 1 right ) ital "mc" rSup { size 8{2} } } {}

When motionless, we have v = 0 size 12{v=0} {} and

γ = 1 1 v 2 c 2 = 1 , size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } =1} {}

so that KE rel = 0 size 12{"KE" rSub { size 8{"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 mv 2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {} . To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for γ size 12{γ} {} at low velocities gives

γ = 1 + 1 2 v 2 c 2 . size 12{γ=1+ { {1} over {2} } { {v rSup { size 8{2} } } over {c rSup { size 8{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 velocity here, most terms are very small. Thus the expression derived for γ size 12{γ} {} here is not exact, but it is a very accurate approximation. Thus, at low velocities,

γ 1 = 1 2 v 2 c 2 . size 12{γ - 1= { {1} over {2} } { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } {}

Entering this into the expression for relativistic kinetic energy gives

KE rel = 1 2 v 2 c 2 mc 2 = 1 2 mv 2 = KE class .

So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when v << c size 12{v"<<"c} {} .

It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that γ size 12{γ} {} becomes infinite as v size 12{v} {} approaches c size 12{c} {} , so that KE rel also becomes infinite as the velocity approaches the speed of light. (See [link] .) 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.

So 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 size 12{c} {} always add to less than c size 12{c} {} . Both the relativistic form for kinetic energy and the ultimate speed limit being c size 12{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.

In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.
This graph of KE rel size 12{"KE" rSub { size 8{"rel"} } } {} versus velocity shows how kinetic energy approaches infinity as velocity approaches the speed of light. It is thus not possible for an object having mass to reach the speed of light. Also shown is KE class size 12{"KE" rSub { size 8{"class"} } } {} , the classical kinetic energy, which is similar to relativistic kinetic energy at low velocities. Note that much more energy is required to reach high velocities than predicted classically.

Comparing kinetic energy: relativistic energy versus classical kinetic energy

An electron has a velocity v = 0 . 990 c size 12{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 size 12{9 "." "11" times "10" rSup { size 8{ - "31"} } " kg"} {} .)

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Practice Key Terms 3

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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