bei48482_FM

Relativity 29

Kinetic Energy at Low Speeds

When the relative speed is small compared with c, the formula for kinetic energy

must reduce to the familiar ^12 m^2 , which has been verified by experiment at such speeds.

Let us see if this is true. The relativistic formula for kinetic energy is

KEmc^2  mc^2 mc^2 (1.20)

Since ^2 c^2 1, we can use the binomial approximation (1 x)n 1 nx, valid

for |x| 1, to obtain

 1  c

Thus we have the result

KE  1  mc^2  mc^2  m^2 c

At low speeds the relativistic expression for the kinetic energy of a moving object

does indeed reduce to the classical one. So far as is known, the correct formulation of

mechanics has its basis in relativity, with classical mechanics representing an approxi-

mation that is valid only when c. Figure 1.16 shows how the kinetic energy of

1



2

^2



c^2

1



2

^2



c^2

1



2

1



 1  ^2  c^2

mc^2



 1  ^2  c^2

Kinetic

energy

Figure 1.16A comparison between the classical and relativistic formulas for the ratio between kinetic

energy KE of a moving body and its rest energy mc^2. At low speeds the two formulas give the same

result, but they diverge at speeds approaching that of light. According to relativistic mechanics, a body

would need an infinite kinetic energy to travel with the speed of light, whereas in classical mechan-

ics it would need only a kinetic energy of half its rest energy to have this speed.

1.4

1.2

1.0

0.8

0.6

0.4

0

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

KE

/mc

2

v/c

0.2

KE = 
mc^2 – mc^2

KE = 12 mv^2

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