As the gas cools, energy is delivered as propulsion (see the sections “Temperature” and “Euler’s equation”). We estimate the propulsion energy in 2 situations:

(1) duct, moving in a straight line;

First, write the conservation of linear momentum


and let delta t --> 0. We get the thrust equation


where Fext is an external force resisting the motion of the system, M_dot is negative because the system is losing mass as it moves and u is the velocity of the exiting gas w/respect to the stationary frame of reference. The second term is known as “thrust”. When u=0 and v=const=c we get an expression for the propulsion energy


(2) rotating duct.

First, write the conservation of angular momentum


and let delt t --> 0. We get the thrust equation


where tau_ext is an external torque due to the resistance of the surroundings; the second term in the Rhs represents rotational thrust; M_dot is negative since the system is losing mass during its motion. Again, if v=const=c the thrust is precisely counterbalanced by the external resistance and the propulsion energy turns out to be the same as above


This formula shows that a gaseous mass m, initially at rest in one frame of reference, makes the transition, at the expense of its own stagnation enthalpy, to a state of rest in another frame of reference, where the 2 reference frames move with respect to each other with constant linear velocity c. Thus, the above formula stems from the physics of a system with variable mass.

If we perform temperature analysis, the exact same formula comes out as the result. The released thrust energy is given through the decrease of total temperature


because, as we saw in the “Temperature” section of this site,


One can call the mc^2 formula an “apparent mass-energy equivalence”, since to an observer in the starting frame where m is initially at rest, it seems that the removal of this mass leads to the liberation of energy. In fact, the mass only transitions between 2 frames of reference and is not equivalent to energy. It releases some of its pressure and thermal energy in the transition.

The information contained in this site is based on the following research articles written by Jeliazko G Polihronov and collaborators: