So a compressed piece of metal whos' acceleration is perfect and at a controlled constant state with absolutely no interferance from the surrounding environment and is actually being supported by the magnetic properties of the world and is staticaly charged with a very powerful charge that will never lose its energy due to the complete redirection of energy flying towards the face of an unfortunate guy is going to happen next right?
The main thing lowering the range of her railgun is friction with the air, to around 50m. With that redirected, the only factor is how long it takes until the coin hits the ground, which, estimating the launch height as 150 cm, we can estimate with:
time to fall = sqrt ( 150 cm * 2 / 980 cm/s^2) = 0.5533 seconds.
At the muzzle velocity stated on the wikia of 1030 m/s, that's a maximum range of 570 meters, or about a third of a mile. Keep in mind this is with an absolutely level shot, no elevation whatsoever.
At 45 degrees, we instead can estimate the range with:
The main problem is aiming that far, as while Accelerator might be able to pull off the necessary calculations, I'm not sure either of them can match an angle to more than a degree or so of accuracy. Acclerator might be able to control his reflection better than that, in which case it would make more sense for Misaka to shoot the railgun straight at him.
Note that the maximum distance a human can see is around 3 miles, so for anything past that they would have to calculate with gps coordinates or something to aim.
Obviously if gravity can be dealt with by Accelerator all of these calculations are right out.
More accurately, the thing limiting her range when shooting coins is because the piece of metal melts from the heat. She gets more range when shooting more heat resistant things. Your range calculations are not taking into consideration the curvature of the Earth. At that velocity, the projectile is going 22m per km away from the surface of the earth at the location of the shot. Since the speed is much less than the escape velocity, the projectile is going to orbit the earth and never touch the ground, as long as there is no air friction.
Good call on me neglecting the curvature of the earth, that's what I get for learning physics in the context of physics engines for games, too used to thinking on infinite flat planes. Though technically I believe the direction of the shot does determine whether the coin would reach orbit or not, the simplest example being shooting a coin straight down.
This also got me thinking, and if my math is correct, the two of them could propel a coin to 54km if shot straight up, which does put shooting down satellites out of reach. Perhaps if they set up a closed orbit such that the coin passed through the rails twice, and thus got extra energy it could be managed, but I would be a bit out of my depth attempting to calculate if that would work, as I'm not familiar enough with either railgun science or orbital mechanics.
