I tried to find a more precise answer but technically an ion-drive is an electric motor, so I'm guessing it's comparable. The thing is that based on the couple demos I watched the flight seems really erratic. It makes me wonder if the fuel supply is somewhat inconsistent i.e. air turbulence makes the actual theoretical energy efficiency moot.
I'm having trouble understanding why it would necessarily be comparable to a traditional electric motor in efficiency. Whether or not it's "technically" an electric motor, it's operating on completely different principles. The first things that come to mind are there are no moving parts to generate friction and heat, and it requires much higher voltage. Wouldn't those aspects alone swing the efficiency one way or the other?
It was a big guess for sure. To be clear, I don't know. I did find this in my search, which suggests it's not exponential[0], so it seems you're correct. There's a drop-off at higher exhaust speeds, but it's unclear to me why that is.
Kinetic energy of the particles (which has to ultimately come out of the battery) is proportional to mv^2. Momentum of the particles (which is ultimately equal and opposite to thrust) is proportional to mv.
