1. Mass energy equivalence
Relation between momentum and energy

2. In quantum mechanics , we considered that kinetic energy could be increased only increasing by its velocity
But now dealing with relativistic mechanics we take mass variation into account

3. Relationship between mass and energy
If force F acting on a particle ,produces a displacement dx then the work done by the force = Fdx the work done must be equal to the gain in the kinetic energy dE of the particle
dE= Fdx
But force being rate of change of linear momentum of the particle , is given by

5. Classical expression

6. Examples for proving equivalence between energy and mass
Nuclear fission
Nuclear fusion
Nuclear reaction processes
Phenomenon of pair production

8. Nuclear fission

10. Example 1
What is the annual loss in the mass of the sun , if the earth receives heat energy approximately 2 cal/cm2/min , The earth sun distance is about 150x106 km
Solution
Rate of energy radiated

12. Example 2
A nucleus of mass m emits a gamma ray of frequency Show that the loss of internal energy by the nucleus is not but is
Solution
The momentum of gamma ray photon is
According to the law of conversation of momentum , the nucleus having mass m will recoil with the momentum in the back ground direction . Therefore, the loss of energy recoiling is

14. Example 3
A certain accelerator produces a beam of neutral K-mesons or kaons mkc2=498 MeV . Consider a kaon that decays in flight into two pions (mpc2=140 MeV)
Show that the kinetic energy of each pion in the special case in which the pions travel parallel or anti parallel to the direction of the kaon beam or 543 MeV and 0.6 Mev

15. Solution The initial relativistic total energy
Ek= K+ mkc2 =325 MeV MeV =823 MeV
Total initial momentum
Total energy for final system consisting of two pions is i

16. Applying conservation of momentum , the final momentum of the two pions system along the beam direction is P1 + P2 and setting this equal to the initial momentum Pk , one obtains
P1c +P2c =Pkc = 655 MeV ii
We have now two equations in the two unknown P1 and P2 , solving we find P1c= 668 MeV or -13 MeV iii

17. Relation between momentum and energy
Relativistic momentum of a particle moving with a velocity v is given by P=mv (1)
Where
(2)
M0 being the rest mass of the particle , from relativity we have E= mc (3)
From 1 and 2 we have

18. Particles with zero rest mass
Photon and Graviton are the familiar examples of particles with zero rest mass . a particle with zero rest mass always moves with the speed of light in vacuum . According to 4 , if m0 =0 , we have E= Pc

19. i.e., a particle with zero mass ( rest mass ) always moves with the speed of light in vacuum . The velcity of the particle observed in some other inertial frame S` is
Where v is the velocity of the frame S` with respect to the frame S in which the velocity of the particle is U, hence U=c we have
Clearly the particle has the same speed c and zero rest mass for all observers in inertial frames.