If you had a loop of electromagnets with an encapsulated payload inside vacuum tubes, how fast could you accelerate the capsule to before it reached some maximum velocity?
What effect would scale have on the answer to that hypothetical?
And would it be worthwhile to have such a loop as a first stage in a mass driver to reduce material and space requirements?
A particle accelerator fits the description given in the OP (albeit for a very small payload) and can reach 99.99....% (more 9s than matters) the speed of light. So that's the maximum speed, essentially the speed of light as long as you're prepared to cope with high $g$ forces - which is well explored in this classic XKCD what-if (also available as a video).$^1$
But raw speed at ground level is not all that useful for getting stuff into orbit. If, through whatever mechanism, you accelerate a payload up to orbital velocity in the low, dense parts of the atmosphere, it will burn up before it gets anywhere. Think about how destructive the forces of re-entry are when a spacecraft is moving at about orbital velocity but going through the very thin upper atmosphere. Moving the same speed at sea level would be many orders of magnitude worse because the air is $10000\times$ denser. Even if that doesn't destroy the payload the air resistance would slow it down vastly, to far below orbital velocity. So you have to get it going even faster initially to compensate, which makes the whole problem worse.
Of course you don't have to accelerate it all the way to orbital velocity, you could "only" give it the speed that the first stage of a standard rocket would, and have the payload include a traditional second stage rocket. This would still leave you having to handle a vastly greater max Q, again, because of the high speed in the thick, dense atmosphere. Which leaves you needing to engineer the whole second stage of a rocket to survive that, as well as building an accelerator that has to accelerator a mass many times larger than the final payload.
Conventional wisdom concluded a long time ago that the atmosphere makes such schemes non starters. That's why some proposals for linear accelerators have them be very long and extend dozens of kilometres in altitude - at that point the air is much thinner and less of a problem. Of course, building such an aerial vacuum tunnel comes with countless difficulties of its own, and we're basically in Space Elevator level of scifi. That's not to say that the general idea wouldn't work somewhere without an atmosphere, like the Moon, or a very thin one, like Mars.
SpinLaunch, however, thinks that the many problems mentioned above are surmountable in the near term. But I'm not the only one who suspects that company will be far more successful at parting investors from their money than getting payloads into space.
$^1$ Pedantry Corner: A particle accelerator works because it's accelerating stuff that has an electrostatic charge - this would not be the case with a mass driver. Electromagnets could be used to give the payload a magnetic field but not an electrostatic charge, so the workings would be fairly different on an engineering level even if Maxwell's Laws are pretty symmetric between magnetic and electric fields. Particle accelerators also have to deal with keeping the particle beam tightly focussed - something which is clearly not an issue when the payload is a single macroscopic object
You can certainly do this, but the maximum potential speed and other technical questions depend heavily on the desired operating parameters of the accelerator. There isn't a simple answer.
While in the loop track, you need a way to provide enough centripetal force to stop the cargo from punching out through the wall of the loop. That could be provided by magnets, or by wheels on a physical track, or whatever -- but also remember that whatever force the loop puts on the cargo, the loop has to withstand an equal and opposite force. The loop forces the cargo onto a curved path, and in return the cargo puts potentially enormous forces on the loop. The physical engineering of the loop determines what the maximums are for that -- the wider the loop, the lower the forces, but you gain less benefit as compared to a straight track.
That curve will also produce equivalent G-loading on the cargo based on the radius of the loop. If you're building a mass driver weapon or delivering the output from a mining/smelting operation, maybe that doesn't matter, because you're firing a solid metal slug and the limiting factor is purely the mechanical forces you're putting on the loop itself. But if you're trying to use a mass driver to launch goods or living humans, you probably need to keep it down to a single digit number of Gs to prevent damaging your products or injuring your passengers.
As an example of this that uses a physical rotor instead of magnets, look at the SpinLaunch project. The technical limitations of the system are non-trivial -- it's not remotely suitable for anything living, and even electronics have to be designed to withstand the forces involved, but I could certainly see a similar system that's intended to operate in a hard vacuum environment and adds a magnetic track to the outbound end of the thing.
Centripetal acceleration, $a_c$ is
$$a_c={v^2 \over r}$$
If 'r' is the radius of the planet, and 'v' is orbital velocity, then '$a_c$' will be 1G. If you travel in a circle with a radius of $0.5 \times r$ but maintain 'v' you'll end up with 2 G's of acceleration. If your payload and passengers can withstand 10 Gs, then the circle's radius will be $r_{planet} \over 10$, or 637.8 km and it's circumference will be ~4000km.
At 10 G's, you can accelerate in a straight line up to orbital velocity in a lot less distance than 4000km. $$t={v \over a} = {7800 \over {10 \times 9.8}} ~= 80sec$$$$d = 0.5at^2 = (0.5)(10 \times 9.8)*(80^2) = 310408m ~= 310 km$$
So, it makes more sense to accelerate in a straight line if your goal is orbital speeds and you want to keep the g-forces in the "human-rated" range.
