Mass Energy Equation

Mass Energy equation :-
       A/C to Netwon's ll low of motion
        
         F=d/dt(mv)
            =m dv/dt+v dm/dt

This force F is displace as body thought a distance dx.
Then increase K.E, i.e dEk of the body equal work done F.dx.
        dEk=F.dx
              =(m dv/dt+v dm/dt).dx
              =m(dx/dt ) dv+v(dx/dt)dm
             =mv dv+v²dm................(1)
     A/C Variation of mass with velocity
         m=m. /√1-(v²/c²)
        m √(c²-v²)/c²=m.
     m²c²-m²v²= m.²c²
   2mdm c²-2mdm v²+2vdv m²=0
       2m(c²dm-v²dm-mvdv)=0
           c²dm=mv dv+v²dm.............(2)
  From eq (1)and(2)
       [dEk=c²dm]......dEk proportional c²dm
 A change in K.E, dEk is diretele proportional to a change mass m.
when a body is at rest. if velocity is zero
               m=m.
when if velocity v is mass because m.
          
i.e  E.K=In(0 to E.K) dEk= c² In(m. to m)dm
       E.K=(m-m.)c²
       E.K=mc²-m.c²
when the body is at rest the internal energy stored in the body is m.c² 
The total energy E of body is the sum of E.K and rest mass energy m.c²
            E=E.K+m.c²
              =mc²-m.c²+m.c²
           [ E=mc² ]  
This eistein mass energy equation.