Einstein mass energy relation discussion and derivation

Einstein’s mass energy relation:

You must have known the relation E = mc2. Let us derive it:

Derivation:

Consider an object of rest mass m’. If force is applied to the object such that it starts moving with relativistic velocity (that is comparable with the speed of light), then its mass will also vary with variation of mass with velocity relation

m = m’/(1-v2/c2)1/2 (1)

Now suppose that work dw will be done due to this force. If the object is displaced along x axis, then work will be:

dw = Fdx

or  dw = (dp/dt)dx      (because from Newton’s 2nd law F = dp/dt)

or  dw = [d(mv)/dt]dx   (because p =mv)

Differentiate R.H.S.

dw = (mdv/dt + vdm/dt)dx    (here m is also a variable quantity, thus m is also differentiated)

or dw = mdvdx/dt + vdmdx/dt

or dw = mvdv + v2dm                                       (2) Continue reading “Einstein mass energy relation discussion and derivation”