Couldn't we, in theory, slingshot the planes into the air to achieve the same relative fuel burn rates ?
I think this only exists on aircraft carriers ?
The required release velocity would need to be so high that drag from air resistance (which increases with the square of velocity) would slow down the vehicle such that there would be no benefit.
However, it then raises the question: What are they doing on aircraft carriers and how is that different ?
https://en.wikipedia.org/wiki/Aircraft_catapult
... I assumed those had some fuel-use benefit, but perhaps their only benefit is shortening the runway ?
You could likely lessen fuel burn of a supersonic aircraft taking off (as you could with a subsonic aircraft) by accelerating it via a catapult of sorts, but you have to do so at acceleration levels tolerable to all passengers, which would make for a long, unwieldy catapult. Release speed is also obviously going to be very subsonic.