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Peter564

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Peter564

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Dale

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You would take the change in energy of the rocket and divide it by the change in energy of the fuel.

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There are several things you might be interested in - are you interested in how much fuel you need to get a specified increase in velocity?

Basically, you need the relativistic rocket equation, and the exhaust velocity of the rocket, if that's what you're trying to measure. I would suggest https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html for the relativistic rocket equation. Wikipedia also has an entry which I'm less familiar with, https://en.wikipedia.org/wiki/Relativistic_rocket.

Other interesting cases you might consider are a relativistic laser driven light sail with perfect transmission efficiency. But I don't recall seeing anything written about that, though it's an interesting case to analyze. Doppler shift of the light from the laser makes the acceleration of the rocket taper off as it reaches higher and higher velocities.

It's really up to you to give more details of what you consider to be "fuel efficiency", one suggestion is the amount of energy in the burned fuel vs the amount of energy that the rocket gains in some specific frame of reference where the rocket's velocity is given by some particular value. You will, of course, find in this case that choosing this definition of "fuel efficiency" makes the "fuel efficiency" dependent on the choice of frame of reference in which it's mentioned.

The key parameters I see are the exhaust velocity of the rocket, the starting mass ratio of the rocket, and the current velocity of the rocket.

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