First, we know that the bullet has the same initial velocity on the moon as it does on the Earth--that is, it exits the gun at the same speed. But as soon as it leaves the gun, it's a different story. First, the moon bullet doesn't have to contend with air resistance--with so little friction, it can maintain its speed longer than the Earth bullet can (it's analogous to shooting a hockey puck across ice, which has very little friction, and shooting across sand, which has a lot of friction. The puck will travel a lot farther on the ice!)
Now, there is the issue of gravity. Assuming your bullet doesn't hit anything (a pretty safe bet on the moon, but don't try this on Earth!) and forgetting about air restistance, the time it takes for the bullet to fall to the ground depends on its initial velocity, the angle at which you shoot it, and the force of gravity.
You can use some basic physics to figure out how far the bullet will fly horizontally and vertically. Say you fire it at some angle "a" (a=0 degrees would correspond to shooting it straight in front of you; 90 degrees corresponds to shooting it straight up). It turns out that the bullet's horizontal range--the total distance it travels before gravity wrestles it to the ground--is given by the equation:
R = v^2 * sin (2a) / g
g is a measure of the strength of gravity. On Earth, it is 9.8 m/s^2. To find g on the moon, we need another equation:
g = G*M/r^2
Now G called the gravitational contstant, M is the mass of the moon, and r is the radius of the moon. Anyway, on the moon,
g = 1.6 m/s^2
So, neglecting air resistance, the bullet will go about 6 times farther on the moon than on Earth. Once you take air resistance into account, the moon bullet has an even bigger advantage!
You might also ask, if the bullet were fired straight up, could it actually escape the moon's gravitational pull and fly off into space? To answer this, we have to compare the moon's "escape velocity" (the minimum velocity an object needs to esscape the moon's gravity) to the bullet's initial velocity. The moon's escape velocity is about 2.38 km/s, but a bullet typically travels at only 1 km/s. So take cover--even in this case, what goes up must come down!
