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Relativistic Kinetic Energy
Relativistic kinetic energy can be expressed as: $E_{k} = \frac{mc^{2}}{\sqrt{1  (v/c)^{2})}}  mc^{2}$ where $m$ is rest mass, $v$ is velocity, $c$ is speed of light.
Learning Objective

Compare classical and relativistic kinetic energies for objects at speeds much less and approaching the speed of light
Key Points

Relativistic kinetic energy equation shows that the energy of an object approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary.

Kinetic energy calculations lead to the massenergy equivalence formula:
$E_{rest} = E_{0} = mc^{2}$ . 
At a low speed (
$v << c$ ), the relativistic kinetic energy may be approximated by the classical kinetic energy. Thus, the total energy can be partitioned into the energy of the rest mass plus the traditional Newtonian kinetic energy at low speeds.
Terms

special relativity
A theory that (neglecting the effects of gravity) reconciles the principle of relativity with the observation that the speed of light is constant in all frames of reference.

Lorentz factor
The factor, used in special relativity, to calculate the degree of time dilation, length contraction and relativistic mass of an object moving relative to an observer.

classical mechanics
All of the physical laws of nature that account for the behaviour of the normal world, but break down when dealing with the very small (see quantum mechanics) or the very fast or very heavy (see relativity).
Full Text
In classical mechanics, the kinetic energy of an object depends on the mass of a body as well as its speed. The kinetic energy is equal to the mass multiplied by the square of the speed, multiplied by the constant 1/2. The equation is given as:
where
The classical kinetic energy of an object is related to its momentum by the equation:
where
If the speed of a body is a significant fraction of of the speed of light, it is necessary to employ special relativity to calculate its kinetic energy.
It is important to know how to apply special relativity to problems with high speed particles.
In special relativity, we must change the expression for linear momentum. Using
Since the kinetic energy of an object is related to its momentum, we intuitively know that the relativistic expression for kinetic energy will also be different from its classical counterpart. Indeed, the relativistic expression for kinetic energy is:
The equation shows that the energy of an object approaches infinity as the velocity
The mathematical byproduct of this calculation is the massenergy equivalence formula (referred to in ). The body at rest must have energy content equal to:
The general expression for the kinetic energy of an object that is not at rest is:
At a low speed (
Thus, the total energy can be partitioned into the energy of the rest mass plus the traditional classical kinetic energy at low speeds.
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Key Term Reference
 energy
 Appears in this related concepts: Surface Tension, Introduction to Work and Energy, and The Role of Energy and Metabolism
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 kinetic
 Appears in this related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Sculpture
 kinetic energy
 Appears in this related concepts: Glancing Collisions, Solid Solubility and Temperature, and Escape Speed
 mass
 Appears in this related concepts: Physical and Chemical Properties of Matter, Mass Spectrometer, and Mass
 momentum
 Appears in this related concepts: The Uncertainty Principle, Rotational Collisions, and Differentiation and Rates of Change in the Natural and Social Sciences
 rest mass
 Appears in this related concepts: Relativistic Energy and Mass, Particle Current, and Ideal Fluids
 speed of light
 Appears in this related concepts: Relativistic Addition of Velocities, Length Contraction, and Time Dilation
 velocity
 Appears in this related concepts: Rolling Without Slipping, Tangent and Velocity Problems, and Applications and ProblemSolving
Sources
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Cite This Source
Source: Boundless. “Relativistic Kinetic Energy.” Boundless Physics. Boundless, 03 Jul. 2014. Retrieved 24 Mar. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/specialrelativity27/relativisticquantities180/relativistickineticenergy6626210/