In this kind of projectile problem, we always ignore air resistance. (At least through high school Physics.) If we tried to include air resistance, that would make the problem hopelessly complicated, when all we're really trying to do is work with the effects of gravity ... separating out what gravity does and what it doesn't do.
Ignoring the effects of air resistance, there is no horizontal force acting on the bullet after it leaves the gun, so its horizontal acceleration is zero, and its horizontal velocity doesn't change. If it left the muzzle with a horizontal velocity of 240m/s, then its horizontal velocity is that same number, from then until it hits something.
(By the way . . . your question says that Brianna fires a gun horizontally, and then it asks how fast does it travel after one second. I took the liberty of addressing the horizontal velocity of the bullet instead of the gun.)
