# Mass-Energy Equivalence

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## What is Mass-Energy Equivalence?

The mass-energy equivalence is an equation derived from Einstein’s theory of special relativity showing the relationship between mass and energy of a body in it’s rest frame. In mass-energy equivalence, the two values mass and energy differ only by a constant and the units of measurement.

The mass-energy equivalence is,

Where E= Equivalent kinetic energy of the object

m= Mass of the object (in kg)

c= Speed of light (3 x 108 m/s)

The equation states that, in the rest position, the energy of a particle is the product of mass and the square of the speed of light.

According to mass-energy equivalence, even though the total mass of a system changes, the total energy and momentum remain constant. The mass-energy relation states that even in a stationary position, every object possesses certain energy. A stationary body does not possess kinetic energy, it only possesses potential energy and probable chemical and thermal energy. As per the field of applied mechanics, the sum of all these energies is smaller than the product of the mass of the object and the square of the speed of light. When an object is at rest having no momentum, the mass and energy results are equivalent and these results can only be differentiated by c2, a constant that indicates the square of the speed of light. Mass-energy equivalence refers to the fact that mass and energy can be converted into one another. Albert Einstein brought this idea into the limelight, but actually, the discovery was not his. The idea of mass-energy equivalence was first put forward by the French polymath Henri Poincare. The mass-energy equivalence arose from special relativity as a paradox described by Henri Poincare.

## Derivation of Mass-Energy Equivalence:

There are two methods to derive the mass-energy equivalence. The first method is the simplest one. Let’s see.

First, consider an object moving approximately at the speed of light. There is a uniform force acting on it, and due to the force, energy and momentum are induced.

Now, the increase in momentum of the object= mass x velocity of the body (as the force is constant)

As we know, E= F x c —> 1

Where E= Energy gaine

F =Force and c= Distance through which the force acts

Now, The momentum gained= Force x Duration through which force acts

We know, Momentum= Mass x Velocity

Hence, The momentum gained= m x c

Therefore, Force= m x c —> 2

Next, combining equation 1 and 2, we have,

An object seems to get heavier whenever it is in speed. The equation shown below indicates the increase in mass due to speed,

Where m= Mass of the object at the travelling speed

m0= Mass of at a stationary position

v= Speed of the object and

c= Speed of the light

An object possess kinetic energy during motion and it is given by,

E= ½ (mv2)

Now, the total energy possessed by the object is approximately equal to kinetic energy and increases in mass due to speed. Hence,

E≅ (mc2) + ½ (mv2)

E- (mc2) = ½ (mv2) , for small v/c

E= Relativistic kinetic energy + mc2

The relativistic kinetic energy depends on the kinetic energy and speed of the object. Now, let’s simplify the equation by making the speed of the object as zero. Hence, the equation becomes,

E= 0+mc2

Therefore, E= mc2

## Applications of Mass-Energy Equivalence:

Einstein’s equation has imperative applications in some specific fields. Let’s see some of these important applications.

• Einstein’s theory is used in the process of nuclear fission and nuclear fusion. A huge amount of energy is generated during fission and fusion reactions. This phenomenon is used in developing nuclear weapons and nuclear power.
• The mass-energy equivalence is used in finding out the binding energy in an atomic nucleus. The binding energy is calculated by measuring the masses of various nuclei and subtracting them from the sum of masses of protons and neutrons. The measurement of binding energy is utilized to formulate the energy released during nuclear reactions.
• The mass-energy equivalence is used to find out the change in mass during the chemical reactions. Breakage and formation of new bonds occur whenever there is a chemical reaction. A change in mass take place during the exchange of molecules. For chemical energy, Einstein’s equation can be written as,

## E= Δm x c2

Where Δm= Change in mass

• The mass-energy equivalence also plays an important role in the radioactivity of various elements. Radioactivity produces X-rays and gamma rays. The same principle is used in many radiotherapy equipments.
• Einstein’s equation is used to understand the universe, it’s constituents, and the age of planets.
• In many surgeries, opening and stitching of body parts will not be done. In such surgeries, cath lab is used. This works on mass-energy equivalence.
• This equation is used to understand the effect of gravity on all stars, moon, and planet.
• It is used to measure the age of fossil fuels

Where E= Equivalent kinetic energy of the object

m= Mass of the object (in kg)

c= Speed of light (3 x 108 m/s)
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