This papers presents an analytical integration for the Ampere's force law. It then makes estimations of the operational characteristics of the railgun based on the Ampere's force law. Operating at a current of 300kA, a 4m long tripled railgun may fire a 1kg projectile reaching an exit speed of 2020m/s (Mac 5.9) with kinetic energy of 2.04MJ. It is estimated that the ohmic loss is just about 3% of the kinetic energy. When the operation of the railgun is analyzed based on the Lorentz magnetic force, there is difficulty in identifying the precise seat of the railgun recoil. In contrast, the analysis done based on the Ampere's force law could precisely specify the seat of recoil of the railgun; it is at the `empty space' in the interface separating the atoms of the rails and the atoms of the gun breech. During firing, contrary to expectation, the rails would be under tension and not compression.