Current projections (that I can find) are a 5 inch projectile at around 16,000 m/s. What's the math on that? (Oh, that's METERS/second, not miles, in case anyone gets confused. Not sure this warning is necessary because that would be 8% of the speed of light.)
The math is beyond me. I know my limits.
Drag is (sorta, at slow speeds) calculated as D = 1/2*p*Cd*A*v^2 where p is the density of air, Cd is the coefficient of Drag, A is the area of the projectile, and v is velocity.
For this problem Density is changing (going down and back up) as the projectile arcs, and velocity will be dropping from the very beginning. SO you can see that the drag acting on the projectile is going to be a moving target. I suspect there are some integrals that could approximate it, or computer models that could crunch a ton of numbers and come close, but both are well beyond me. Birdman would have to chime in.
I'm just spitballing that with a velocity that high, drag is going to do all sorts of weird things to slow that projectile down. Look at all the energy that is being used to ignite the air behind that thing. for all that they put 32 megajoules into acceleration it's dumping energy fast. Once it gets under 1000 or so m/s it's just a normal 5" gun with no explosives.
I could be really off, but 250 miles is a LONG way through air with no boost.