## 29   The Kinetic Share of E'

The calculation below is based on the following equations derived in 22 to 28 :

• E  =  M · c2  +  U  +  P · V
• E'  =  E / √  =  M · c2 / √  +  U / √   +  P · V / √
• E'  =  M · c2  +  U' P' · V' Ekin'   =  M · c2  +  U · √  +  P · V · √  +  Ekin'

We get total kinetic energy  Ekinas follows:

Ekin'   =  E'  –  M · c2  –  U'  –  P' · V'   =

mmmjE / √  –  M · c2  –  U · √   –  P · V · √   =

mmmj= (  M · c2  +  U  +  P · V  ) / √  –  M · c2  –  U · √   –  P · V · √   =

mmmjM · c2 · ( 1 / √  – 1 )  + ( U  +  P · V  ) · ( 1 / √  – √ )  =

mmmjM · c2 · ( 1 / √  – 1 )  + ( U  +  P · V ) · ( 1 / √ ) · v2 / c2   =

mmmM · c2 · ( 1 / √  – 1 )  + ( U'  +  P' · V' ) · v2 / ( c2 v2 )

The first term is the well-known expression for the kinetic energy of the fast mass M. The second term represents the kinetic energy due to the transport of internal energy U'  and the transport of thermodynamic work  P' · V'  . If we add to the above expression for Ekin'  the terms  M · c2  +  U' P' · V' , we should not be surprised to get  E'  again. The calculation is straightforward.