Brianna fires a gun horizontally from a height of 500 m and with a muzzle velocity of 240 m/s. how fast does it travel horizontally after one second?

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.)