Your calculation is very very incorrect, not the math, but the concept you are missing too many variables.
The US navy actually is building railguns, their efficiency is very very low due to the resistance from inductance it seems that if we go the the equations for railguns your 200KM EM gun cannot be built.
Overall the US is designing a 64MJ railgun, this gun can't put anything into orbit, it will have a range of about 20 miles, the ship that is going to be equipped with it is going to have a 78MW power plant and while it can power a single rail gun it will not be able to power multiple ones.
By the US Navy's own calculations it would require 28MW to launch a projectile at 32MJ which which means yeah.... these figures are all off by orders of magnitude.
It seems there is much more to railguns than classical mechanics.
I believe the barreled design to be fatally flawed. An earthquake or other disruption during launch could be catastrophic to the vehicle. Much better for the vehicle to ride next to the track; in failure scenarios, the vehicle can simply detach and glide back to earth.
Yes, it means you have to go through the atmosphere, but it doesn't take far to clear. That will effect the calculations some, but not much. What will really effect the calculations though, is the cost of electricity. Generation costs will surely drop below $0.04.
Further, transmission loss can be almost entirely mitigated by generating and supplying the required power on-track.
Launches in favorable conditions (moon and/or planet alignments) would probably make some launches even cheaper. Of course, if the launcher were operating continuously, the savings would be used up during unfavorable conditions.
You're suggesting that we should magnetically levitate the launch vehicle in free air above a track running along the ground? That seems like it makes the problem a lot harder — instead of having to push it through just the air between here and space, which is about ten tonnes per cross-sectional square meter, you're pushing it through another two or three hundred kilometers of air, which is about another 200 tonnes of air per square meter. You know that shroud of plasma surrounding a re-entering spacecraft? That's the power required to push an orbital-speed object through air — but in that case without even maintaining velocity, let alone rapidly accelerating, and in that case it's the rarefied air of the stratosphere. You're proposing to do that for the majority of the track. That seems like a bad idea.
Yes, generation costs will likely drop significantly below US$0.04/kWh eventually. But that's a Kardashev-Type-1 kind of event. Generating the power on-track may not turn out to be less expensive than long-distance transmission, because it depends on things like sunlight availability. Of the few suitable sites, most are pretty cloudy on one side.
No moon or planet alignments significantly reduce the energy barrier to get to orbit.
jsprogrammer has the merit of having actually done some calculation, which you evidently have not — although if you think resistance is a thing that inductance can have, maybe you are not in a position to do any calculations. Either way, you should rethink your attitude.
Naval railguns don't have the luxury of accelerating over hundreds of kilometers; they are optimized for muzzle velocity, not efficiency. The things I've seen videos of launch projectiles at about Mach 7.5 (2500 m/s) over about 7 meters. That implies an average acceleration of at least 45000 gees, which means that you're going to have to accept significant inefficiencies that you can avoid in a design that accelerates three thousand times more gently.
In practice, both rotary and linear electric motors typically have efficiencies of over 80%, often over 95%. The proposal in question is a 200-km-long linear electric motor running up the side of a mountain. It's clearly feasible, but it won't cost pennies per launch.
You do need to partially evacuate the launch tube to shove your launch vehicle through hundreds of kilometers of it.
It is not plausible that a naval railgun uses only 28 megawatts. Traveling 7 meters at an average of Mach 3.75 takes 5.6 ms; if the total energy output is 64 MJ, that's an average of 11.3 gigawatts†, which is a lot more than 28 megawatts or for that matter 78 megawatts. So what you do is you charge a big low-ESR capacitor bank (at less than 78 megawatts) before the shot, then discharge it during the shot (at tens of gigawatts). If you're doing that, though, you have no limit on how many railguns you can run from your 78 MW power plant, just a limit on how many total shots per second you can fire among all of them. Your entire paragraph on the topic is incoherent nonsense.
For comparison, a .22 LR 30-grain (1.94 g) copper-plated hollowpoint bullet traveling at 500 m/s out of a 510 mm AR-15 barrel only has at most 2 ms to accelerate to its final 240 J energy and therefore requires over 120 kW of power.
Classical mechanics are perfectly adequate for all of this. No relativistic or quantum effects are relevant. Your suggestion otherwise is absurd.
Calculations in units(1) format for those who want to check them:
† With constant acceleration the power output ramps up linearly and ends up at twice the average. With constant power the acceleration ramps down instead, which means that you have to start out at even higher accelerations to get the same average acceleration, and your total time in the barrel is shorter, so your average power is higher, although I don't feel like doing the simple calculus to quantify this at the moment. In either case you have at least a point where the power is a few times higher than this average.
