You mean these links? Is there a part that deals with this specific problem, or do I need to read it all?
I’d advise reading it all, it shouldn’t take long, it looks like more than it is. I’d advise reading lessons one and two, that should be sufficient, if you don’t understand just keep going.
Edit: In fact, reading Up to Lesson 2 section e and just forgoing F should be sufficient.
Nah! I’m not that bothered! 
There is one thing I’d still like to point out though, when you ask “why” you’re pretty much asking “what is gravity and how does it work” to which we have to reply “we have no clue.” So the fact that they drop at the same rate is because it works that way. We can show you the math, but it’s already assuming (and proven) that gravity has the effect that, well, that gravity has.
What he said.
Again, what part is missing? This is basic, basic physics. What force could act on the bullet in a vacuum that would prevent it from falling?
He’s basically asking how gravity works, without actually knowing he’s asking that. 
Isn’t momentum what keeps planets in orbit? And stops them falling prey to gravity? And doesn’t a fired projectile have more momentum than a dropped one, hence reducing it’s susceptibilty to gravity. Until the momentum drops sufficiently enough for it to drop?
No I’m not. I’m asking how it works equally effectively on a moving and a motionless body. ie. a bullet.
Okay, I think you’re getting confused using terms like “prevent.” All forces are equal, one doesn’t “prevent” another from happening. You don’t rule out any forces, at the end of the equation you just sum everything up in the horizontal direction, and then in the vertical. Moving forward has NO BEARING on moving down, even when taking into account aerodynamics (it’s still an upward/downward force). I can’t really think of a good way to explain this. Maybe I shouldn’t have said gravity, you’re effectively asking how PHYSICS works. The explanation for the problem is that “gravity and the forward motion don’t effect each other” we can’t honestly answer “why.”
And as for orbit, it’d be much easier to explain that after you understand this problem.
Nothing stops anything from falling prey to gravity. Planets in orbit are, in fact, falling continually. The reason that planets don’t crash into the Sun is that, on that scale, gravity isn’t uniform: When a planet’s on the east side of the Sun, it’s falling west, and when it’s on the west side of the Sun, it’s falling east, and it all cancels out in the end.
However, on the scale that most guns and bullets work, gravity is (to a very good approximation) uniform. If we’re not worried about the curvature of the Earth, then we’re not worried about the nonuniformity of gravity.
Just so we are on the same page, can we assume that we are talking about a performance in a vacuum and on a flat surface? I realize the OP had something else in mind, but let’s get our simple physics straight first. You are saying that two objects dropped from the same height will take different times to reach the ground? That’ll be news to Galileo, not to mention your high school physics teacher.
Other than intuition, can you produce any evidence that contradicts these equations for falling bodies?
Link:
That’s pretty much what Galileo postulated and Newton calculated, and except for Einstein’s modifications, the way physics works today.
Galileo didn’t fire the cannonball as he dropped his feather, did he?
No. Gravity takes no holiday.
Now if you are thinking that if a bullet could be fired fast enough, it would either escape Earth’s gravity or go into orbit, you are right. Such a scheme has been proposed (a rail gun launcher) to put objects into orbit, and if they can withstand the extreme acceleration (human bodies probably couldn’t), there’s no reason why that wouldn’t work. But this isn’t because the projectile has “reduced its susceptibility to gravity,” but because Earth’s gravity becomes a smaller factor when extreme acceleration is involved. Earth still pulls on the object, just like a dropped bullet, but the pull is less significant because the Earth’s surface is dropping away faster than the object is falling. In fact, that’s a perfect description of a satellite – it drops at the same rate as the ground moves away from it; it is continually falling.
NO! I am saying one is dropped, and one is fired, at a velocity that is not going to escape the earths gravitational pull.
Two objects dropped from the same height is the exact same property as firing one. The only difference is the fired one also moves forward.
So, if it moves forward fast enough, woudn’t this delay it’s downward motion even slightly? It just doesn’t make sense that a bullet fired from a pistol or rifle barrel would land at the exact same speed as one that just fell out of the barrel?
Planets remain in orbit due to the cancellation of the instantaneous radial component of the object’s momentum (using the central mass as our coordinate frame) and the application of gravitational force, leaving only the tangential vector of momentum for a perfectly circular orbit. The orbits of natural bodies are never quite circular, and some are actually elliptical, which means that they’ll speed up in some parts of their orbits and slow down in others as described by Kepler’s Second Law of Planetary Motion (“The line from the Sun to a planet sweeps out equal areas for equal periods of time.”) The orientation of the vectors obviously changes as the orbiting body moves through its path; for a stable (elliptical) orbit, this means that the resultant component is always pointing over the horizon. As Chronos notes, this means that the object is perpetually falling around the central mass. Gravity is acting on it at all times; we know this because without the operation of gravity the object would proceed in a straight line with a speed proportional to its momentum.
As for the bullet question, both bullets will fall down at the same speed provided that neither starts out with a vertical component, i.e. someone doesn’t throw the unfired bullet down or point the barrel upward. However, as a practical matter, sights are typically set so that the barrel is actually pointed slightly upward, giving the bullet a very gentle arcing trajectory that Crafter_Man speaks of. This is in order to give a point of aim that is “flat” with respect to the shooter out at 100-300 meters (for rifles, depending on caliber and application) which makes it easier to accurately account for up/down grade and windage corrections. This can be seen with exaggerated effect with black powder cartridge silhouette shooting, where the distance of the target (600-1000 meters) and the slowness of the bullet require a discernible up angle and arc. This slight up angle will impart tiny degree of lift (due to the asymmetry from the bullet’s angle of attack and exchange of momentum for lift) but I doubt it is even physically measurable without very sophisticated equipment, and probably lost in other errors for any practical experiment.
The spinning of the bullet (imparted by the rifling for gyroscopic stability in flight) does not add any lift, and in fact if anything probably reduces the effectiveness of any lifting impulse. This notion has probably been conflated with the lifting force imparted on a golf ball via backspin which is caused by the Magnus effect; in this case, however, the lift comes from the spin of the ball on the horizontal axis perpendicular to the direction of flight. (This is also what causes balls to arch left or right; same for curved trajectories in baseball and soccer.) At all times gravity acts on objects in exactly the same way, assuming a constant planar gravitational field.
In essence, the same forces that retard the falling of the dropped bullet will also act on the fired bullet, and independent of the drag forces which occur due to the bullet’s ‘forward’ (i.e. horizontal) motion.
Stranger
There is no ‘delay’; gravity (in a constant field) acts on all objects equally and all the time, even those that aren’t actually in motion. The fired bullet will travel further before hitting the ground if you give it more forward velocity, but it will fall at the same rate as any other bullet (if we assume it to be fired at an initial horizontal trajectory) whether fired or dropped.
Stranger
No.
It may not make sense, but it’s the foundation of modern physics.
I have scanned a page from my high school physics book, Physics, by Heath, published by the PSSC (Physical Science Study Committee), illustrating how falling bodies work regardless of forward motion or lack of it. Please forgive the crappy quality of the scan – I had trouble getting it squared off on the scanner. Just believe that the nearly-horizontal lines are truly horizontal in the experiment.
Picture with caption. The text states, “We see that the vertical motions of the two balls are identical despite the fact that the horizontal motions differ…The presence of the downward force does not change the horizontal motion; and the existence of horizontal motion does not change the effect of the downward force on the vertical motion. Our observation shows that the horizontal and vertical motions are independent…the vertical motion is the same as free fall…”
Now if you say that a gently pushed ball is different from a rapidly-fired bullet, I am going to say, “Why, how and wherefore?”