Mass driver, on the earth ground, is not necessarily a great idea, but there are still some ongoing projects from the gun to space and they may make sense, so let's crunch some numbers.
Launch energy to spend is
40MJ/kg is a reasonable number, this number represents kinetic energy at 8944 m/s. Going through the air for small, under ton projectiles, if I remember correctly from that ramjet gun project - losses can be up to 50%. For bigger projectiles situations tend to improve, the bigger(decreasing proportion of surface area/mass) the projectile, the lesser the losses, longer in air shallower the angle more the losses. But, let's go with 50% losses with that aspect, maybe too much but it depends on the specific trajectory, angle of ascent.
Storage losses get efficiency 70-80 percent - batteries, capacitors, mechanical storages + plus some transmission losses - 70 percent storing efficiency.
The track itself isn't an ideal machine, whatever happening there - self-cleaning procedures, straightening devices, cooling coils or conditioning the track or whatever - let's go 50%.
So we may end with a 17.5 percent efficiency and that 40 MJ/kg bloats to about 230 MJ/kg and total 4600 GJ or 1.26 GWh. Let's round it up a little, as you may need some additional sled(like in rocket sled) that does not go in space but keep the thing on track and say it 1.5 GWh(5.4 TJ).
So, launch parameters end up be:
- 12.6 km/s velocity (insane), 57g (a lot), 140km acceleration track (short, not human-rated, kinda)
total energy to produce and to charge for launch - 1.5 GWh
price with 0.1 bucks per kWh - 150 grand.
- 30% loss in charge-discharge, 50% loss in track this way in capacitor banks it needs to store about 2 times more than it needs in an ideal case and again 2x due we need to shoot faster than if there would be no air - total 4x that of an ideal case
- and overall we need to supply from power plant to the system 5.7x than of an ideal case, a total of 1.26(1.5) GWh
- Low Earth orbit (LEO) Trans-Lunar Injection (TLI) 3.20km/s delta-v, this way you need about 3km/s extra, and it is around +57% in terms of energy.
150MW, ITER-III - yep, pretty much typical ITER - 840m3 plasma volume, 500 MW heat with a big building and all that around.
And from the total, we see it may be sufficient to make 1-2 launches a day - so as a power source - it is okay.
more frequent launches and you may need proportionally more of those or get decommissioned nuclear plant - just one block gets you about 10 times more, and there are few of those in a plant. (or just pay electricity bills, not that expensive)
- even a diesel(methane) powered generator will do, it definitely a more efficient solution than using that fuel in a chem rocket.
Last km's, peak power consumption is around 2800 GW (57g, top speed 12.6km/s, x2(due loss 50% in the track), 20000kg payload).
This is x40 of peak power of Germany(?) electricity production or about 242GW (average), for about 22.1 seconds, yeah a bit too much, for the most ways to generate energy - but, for a fusion reactor, it may be actually not that bad.
Not necessarily for Tokamak ITER - but in general, for fusion as it is now - they do work for a short time, as for now and the launch needs just a 22 seconds - so maybe exactly the necessity for such burst power is what makes it a good match to that time fusion technologies - they do not need to work too long, they can work in pulse mode as it exactly what the track may need. in some circumstances may be a match made in heaven, with some MHD generator, which in the mature stage may allow launch payloads with seconds intervals - fusion-powered rocket launches, how cool is that definitely something to look forward to.
this fusion direction requires some further investigation. it does not have to sustain the plasma, but work more like a prototype of a fusion rocket engine, be more open architecture, not tokamak, fewer stability issues(maybe) as they are blown out as exhaust.
it requires about 18'000 ITER's like in the q to power the thing directly if they can't give proper peak power, and if they can follow power profile in those 22 seconds then it may be 1600 of those blocks
with charging, once an hour launch requires about a 1.5 GW power line, which isn't that much. Do not get it wrong, it's a good meaty line, but power plants can be 1000's km away.
fun fact first - per each meter of the track it needs an equal amount of energy. So energy storage may be a line of the same boxes along the track and in that sense, its volume does not matter that much, mass does as it correlates more directly to the price of a solution.
Thus we get our 1.5 GWh (5.4 TJ) and accounting our assumed 70% losses in charging(a bit wrong, but...), spec capacity per each meter of the track has to be 27 MJ (btw, a few kg's of propane, 49.6MJ/kg - can be an option, similar to EMP charge).
From wiki situation looks similar for Lithium batteries and Flywheels (good stuff btw) - about 57 kg of the stuff per meter of track. (output discharge is too low, however, so it is just a number for energy-storing)
specific energy is how much energy they can store, mass-wise. specific power - how fast or how powerful is the discharge
So energy-wise it needs about 1500 kg of super caps(18000J/kg, 5Wh/kg), but discharge wise it is a bit tricky.
with 10kW/kg specific power and total mass 210'000 tons of those super caps, total discharge power is 2100 GW, which is less than peak power of 2800 GW it needs more of those capacitors 280'000 tons or 2000 kg/m, realistically probably even more, but we got quite good safety margin previously with all efficiencies and losses, time to use it here.
- conveniently the table contains numbers for more regular capacitors, about the same for them, 10 times better discharge power and 10 times worse in terms of energy-storing, but if they are 10 times cheaper it can be an alternative.
As a volume, let's take LSG/manganese dioxide with 42 Wh/L(150kJ/L) with 10kW/L discharge(which was used) and 10'000 cycles (suspicious dog.jpg, not in this setup, lol)
And we get 240L per meter of track or 12 of 5-gallon buckets(i guess).
In total, we get 33600 cubic meters, a lot, but a station each km will be 240 cubic meters per station, with spacing like few meters high 10x20m rectangle.
problems and non problems
@thewildnobody in his answer addressed them well enough.
However, there are still things to mention, maybe even positive ones.
NACA, an agency that predated NASA did some testings back in the days and submersive half-backed setup was used for testing and it did allow some testers to endure 30g(or something like that) for about 30 seconds - limits were due to centrifuge specs and holding breath limits of a tester. And max for short time was like 83g(not with the setup) - so in that sense, 57g aren't impossible, even if problematic they are.
And due to the atmosphere you have to be prepared for high g's, even if you have a longer track with moderate 5g as you wish it, then still there will be few seconds of declaration after the exit from massdriver, and it does not make it easier than it will be an opposite force, and this acceleration may be very high, in a few seconds, it may drop few km/s sooo, losses due to interaction with the air may be quite high, so the deacceleration. For some types of non-manned vehicles, it is still fine - but for humans at any rate you may need a good jelly box. All the viability depends on how good that jelly box is.
excess of jelly or water probably can be used for some ablative solution for those critical few seconds before the craft leaves the dense atmosphere, but it is not certain.
p.s. for some reason I stuck with numbers and made plenty of mistakes, now I believe it is more or less okay, but still, it is more or less an example of calculations.