According to classical mechanics, force F is given by the time derivative of the product of
mass (m) and velocity (v), i.e., momentum of the body, and is given by
f = d /dt (mv)
However, accordin0 to the theory of relativity, both m and v are variable
So , f d/dt (mv) =mdv/dt +v dm/dt
When a particle is displaced through a distance ‘dx’ on application of a force F, then the
increase in kinetic energy is given by,
dk =f x dx
= [m dv/dt +v dm/dt ] x dx/dt x dt
= [m dv /dt +v dm/dt] x v x dt
Dk =v2 dm +mv dv
Now, we know that
è m2 = m02 c2 /c2 –v2
ð M2 c2 -m2v2 = m02 c2
On differentiation both sides, we get,
2m x dm x c2- [2m x dm x v2 + 2v x dv x m2] = 0
ð 2 m x dm x c2 = 2m [ dm x v2 + v x m x dv]
c2 dm = v2 dm + v m dv
By comparing equations (2) and (3), we have,
D K = c2 d m … (4)
Now, we can consider that the body initially at rest and on application of force it acquires a velocity v. Also the mass of the body increases from m0 tom. Thereby, the total kinetic energy acquired by the body is given by.
ʃ dk = ʃ m mo c2 dm
k =c2 (m m0)
This is the increase in kinetic energy due to the increase in mass.
Now, the total energy of a moving body is the total of kinetic energy due to motion and the energy at rest.
So, E = k + m0 c2 = c2 (m- m0) + m0 c2 = m c2
Hence, [E=m c2 ]
This equation gives the universal equivalence between mass and energy. This is another major contribution of Einstein. It signifies the manifestation of energy into the matter and vice-versa. Relativity shows that mass is not any independent quantity, however, it is a form of energy. Its experimental evidences are well known in nuclear fission and fusion reactions. The conversion of mass and energy and the reverse are also illustrated in nature itself. When a particle collides with an anti-particle there is a mutual annihilation and the total mass is converted into radiant energy. Thus, the conservation of mass takes place into energy. Other way, when a radiant energy comes near a charged nucleus, particle and anti-particle are created. Thus the energy is converted into mass.