Swat, this 500 mile range for your big gun as opposed to 300 miles needs to be put into perspective.
This proposition increases the range by over twice the maximum range of the Paris Gun, which had the longest range of any artillery ever. Look at the map Wikipedia has on the Paris Gun page. To put that in perspective again, a modern railgun has a range of a bit over 200 miles, so this proposed range hike increases said range by that of a railgun. I can appreciate your points (backed up by mathematics or no), but the range of the Longbow will only be allowed the 300 miles which we compromised on, a number you suggested.
EDIT: I actually did some algebra based physics and given known information on the Longbow (V0 = mach 7 = 2382.03 m/s) and assuming a 45 degree angle of elevation and not factoring in air resistance or the deployable wings (which, by the way, make little sense given that firing a railgun produces electromagnetic destruction, so the projectile should be incapable of getting any signals to do so, and by rights I think that deploying the wings would interfere with the trajectory of the gun given that it would produce some jolt. If you could do it manually, even barring the jolt, the only feasible way I can think of would be to have the wings swept forward which lends itself to a decrease in lift.), the longbow should have a maximum range of a bit under 360 miles. Take it or leave it, but I'd say that not factoring in air resistance is fairly generous considering that I don't think that the deployable wings are feasible (If you feel otherwise please provide an existing counterexample or evidence of some sort).
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