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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