In the special
theory of relativity, there is a new expression for kinetic energy. The difference between the relativistic expression and the
classical expression does not come from any conceptual difference in the idea of
energy. Kinetic energy still is the amount of work required to move an object.
To derive the expression for relativistic kinetic energy you firstly define force, which is defined as the change of momentum over time.
F ≡ dp/dt
Relativistic
momentum is given by the expression
p = γm
v. Where γ represents the
Lorentz Transformation gamma factor.
γ = (1-v2⁄c2)-1/2, c represents the speed of light in a vacuum.
Substituting relativistic momentum into the equation which defines
force gives:
F = d(γmv)/dt
Kinetic energy is correctly defined in the write-up by Blush Response above, it is:
Ek = ∫Fds = ∫d(γmv)•ds/dt = ∫γmvdv
Leaving out some of the complicated
algebra, the solution to the relativistic exrpession for kinetic energy is:
Ek = mc2(γ - 1)
Expanding γ with the
binomial theorem gives:
γ = 1 + 0.5•
v2⁄
c2 + ...
When substituting into the relativistic equation of kinetic energy, you approach the classical result 0.5•mv2 when speeds are much smaller than the speed of light.
At large velocities the differences are significant. Classically if you are travelling at 80% the speed of light, and increase to 95% the speed of light, it requires an increase of 41% more energy. Relativistically, to increase from 80% of the speed of light to 95% of the speed of light, it requires 92% more energy, over twice as much as predicted by the classical theory.