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The relativistic energy–momentum equation holds for all particles, even for massless particles for which m 0 = 0. In this case: = When substituted into Ev = c 2 p, this gives v = c: massless particles (such as photons) always travel at the speed of light.
Total energy is the sum of rest energy = and relativistic kinetic energy: = = + Invariant mass is mass measured in a center-of-momentum frame. For bodies or systems with zero momentum, it simplifies to the mass–energy equation E 0 = m 0 c 2 {\displaystyle E_{0}=m_{0}c^{2}} , where total energy in this case is equal to rest energy.
For example, for a speed of 10 km/s (22,000 mph) the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg). The relativistic relation between kinetic energy and momentum is given by
Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including rest mass and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term.
These two types of relativistic particles are remarked as massless and massive, respectively. In experiments, massive particles are relativistic when their kinetic energy is comparable to or greater than the energy = corresponding to their rest mass. In other words, a massive particle is relativistic when its total mass-energy is at least twice ...
This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case θ = 0, and cos θ = 1, which gives:
The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation.
Kinetic energy in special relativity and Newtonian mechanics. Relativistic kinetic energy increases to infinity when approaching the speed of light, thus no massive body can reach this speed. Tests of relativistic energy and momentum are aimed at measuring the relativistic expressions for energy, momentum, and mass.
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