Relativistic Kinetic Energy
Relativistic kinetic energy can be expressed as:
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

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).

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.
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:
Time Magazine  July 1, 1946
The popular connection between Einstein, E = mc2, and the atomic bomb was prominently indicated on the cover of Time magazine (July 1946) by the writing of the equation on the mushroom cloud itself.
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.
Key Term Reference
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Postmodernist Sculpture
 kinetic energy
 Appears in these related concepts: Solid Solubility and Temperature, Pressure, and Types of Energy
 mass
 Appears in these related concepts: Mass Spectrometer, Pop Art, and Mass
 momentum
 Appears in these related concepts: Inelastic Collisions in Multiple Dimensions, The Second Law: Force and Acceleration, and Differentiation and Rates of Change in the Natural and Social Sciences
 rest mass
 Appears in these related concepts: Photon Interactions and Pair Production, Relativistic Energy and Mass, and XRays and the Compton Effect
 speed of light
 Appears in these related concepts: Relativistic Addition of Velocities, Length Contraction, and Time Dilation
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
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