# Does increasing the energy of a relativistic particle increase thrust?

I know this particular chestnut has been asked before, but I have a specific variation on the theme:

Particle accelerator space thrust. Specifically, cyclotrons to generate the thrust. They are ideal in their simple robustness and the fact that as long as you provide the square-wave power and appropriate material to the centre, it will continue to accelerate it.

My question is: Accelerating a particle to relativistic speed begins to add energy to it beyond the speed of the actual particle. Would this extra energy provide any increase in thrust if used as an exhaust? What point does the energy put into the system become no longer worthwhile? I've seen some graphs that show the divergence is not significant until ~0.6 or 0.7c, at this point how much acceleration is 'wasted' in increasing the energy of the particle? Can a cyclotron even accelerate particles to that speed? Is there a hard limit on the 'feed speed' for the device i'm not aware of?

The force is given by:

$$F = -\frac{dp}{dt}$$

where $p$ is the momentum of the particles being ejected. The momentum is given by:

$$p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}}$$

And the momentum is related to the energy by the relativistic equation for the total energy:

$$E^2 = p^2c^2 + m^2c^4$$

so:

$$p = \frac{\sqrt{E^2 - m^2c^4}}{c}$$

The point of all this is that even though the velocity $v$ is increasing by only incremental amounts near to the speed of light the momentum continues to increase as you increase the energy. So yes, increasing the energy of your cyclotron will increase the thrust even when the particle speed is very close to $c$.