# Derive the equation of mass energy relationship

# Derive the equation of mass energy relationship

## 4 Answers

^{2}is derived from the equation for kinetic energy Ke = mv

^{2}. The mathematics and concepts of special and general relativity shows that the absolute maximum velocity anything can have is the speed of light. The maximum amount of energy anything can possess is simply calculated from its mass and this maximum velocity squared.

In physics – in particular, special and general relativity – **mass–energy equivalence** is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy; energy is a property of all mass; and the two properties are connected by a constant. This means (for example) that the total internal energy *E* of a body at rest is equal to the product of its rest mass *m* and a suitable conversion factor to transform from units of mass to units of energy. Albert Einstein proposed mass–energy equivalence in 1905 in one of his *Annus Mirabilis* papers entitled "Does the inertia of an object depend upon its energy-content?"^{[1]} The equivalence is described by the famous equation:

where *E* is energy, *m* is mass, and *c* is the velocity of light. The formula is dimensionally consistent and does not depend on any specific system of measurement units. The equation *E* = *mc*^{2} indicates that energy always exhibits relativistic mass in whatever form the energy takes.^{[2]} Mass–energy equivalence does not imply that mass may be "converted" to energy, but it allows for *matter* to be converted to energy. *Mass* remains conserved (i.e., the quantity of mass remains constant), since it is a property of matter and also any type of energy. Energy is also conserved. In physics, mass must be differentiated from matter. Matter, when seen as certain types of particles, *can* be created and destroyed (as in particle annihilation or creation), but a closed system of precursors and products of such reactions, as a whole, retain both the original mass and energy throughout the reaction.

When the system is not closed, and energy is *removed* from a system (for example in nuclear fission or nuclear fusion), some mass is always removed along with the energy, according to their equivalence where one always accompanies the other. This energy thus is associated with the missing mass, and this mass will be added to any other system which absorbs the removed energy. In this situation *E* = *mc*^{2} can be used to calculate how much mass goes along with the removed energy. It also tells how much mass will be added to any system which later absorbs this energy. This was the original use of the equation when derived by Einstein.

*E* = *mc*^{2} has sometimes been used as an explanation for the origin of energy in nuclear processes, but mass–energy equivalence does not explain the origin of such energies. Instead, this relationship merely indicates that the large amounts of energy released in such reactions may exhibit enough mass that the mass-loss may be measured, when the released energy (and its mass) have been removed from the system. For example, the loss of mass to an atom and a neutron as a result of the capture of the neutron, and the production of a gamma ray, has been used to test mass-energy equivalence to high precision, as the energy of the gamma ray may be compared with the mass defect after capture. In 2005, these were found to agree to 0.0004%, the most precise test of the equivalence of mass and energy to date. This test was performed in the World Year of Physics 2005, a centennial celebration of Einstein''s achievements in 1905.^{[3]}

Einstein was not the first to propose a mass–energy relationship (see the History section). However, Einstein was the first scientist to propose the *E* = *mc*^{2} formula and the first to interpret mass–energy equivalence as a fundamental principle that follows from the relativistic symmetries of space and time.

The simple equation *E* = *mc*² is not generally applicable to all these types of mass and energy, except in the special case that the total additive momentum is zero for the system under consideration. In such a case, which is always guaranteed when observing the system from either its center of mass frame or its center of momentum frame, *E* = *mc*² is always true for any type of mass and energy that are chosen. Thus, for example, in the center of mass frame, the total energy of an object or system is equal to its rest mass times *c*², a useful equality. This is the relationship used for the container of gas in the previous example. It is *not* true in other reference frames where the center of mass is in motion. In these systems or for such an object, its total energy will depend on both its rest (or invariant) mass, and also its (total) momentum.^{[12]}

In inertial reference frames other than the rest frame or center of mass frame, the equation *E* = *mc*² remains true if the energy is the relativistic energy *and* the mass the relativistic mass. It is also correct if the energy is the rest or invariant energy (also the minimum energy), *and* the mass is the rest mass, or the invariant mass. However, connection of the **total or relativistic energy** (**E _{r}**) with the

**rest or invariant mass**(

**m**) requires consideration of the system total momentum, in systems and reference frames where the total momentum has a non-zero value. The formula then required to connect the two different kinds of mass and energy, is the extended version of Einstein''s equation, called the relativistic energy–momentum relation:

_{0}^{[13]}