We have just learned that gravity isn't picky.  Any airborne object is acted upon by it.
When you visited the projectile program mentioned in the previous page, did you notice anything about the difference between the X and Y components?

The falling object is accelerating downward, and moving at a constant speed sideways.  How can you tell?  As the yellow dots progress down the page with the green ones, they get farther and farther apart.
As the red dots progress across the page, they occur at fixed intervals.

THIS should make sense.  Once the object is launched, how many forces are acting on it?  Only one: gravity.  Which way does gravity act?  DOWN.  This is why the object accelerates only downward and not across.       (Remember, we are ignoring air resitance)
Beware!  A projectile launched sideways IS still affected by gravity, and does sink just like a dropped object.  Rifle bullets just seem to defy this because they are going so fast sideways.
This idea extends to satellites: they are falling all the time, but fail to hit Earth because we put them in space with a really high sideways velocity.  This is why astronauts feel weightless!  More about this later!
 

What's the big idea?

HORIZONTAL MOTION IS INDEPENDENTOF VERTICAL MOTION.

So, for a projectile, we can use the kinematic equations involving acceleration to find the height, AND we can use the equation for constant speed to find the distance sideways.
If you follow the shape traced out by a projectile, it makes a parabola.  These sites have good information on parabolas, 3 of these links go to animations, and one goes to a math-type page... #1, #2, #3, #4

By the way, by this time physics books have begun to write "g" in place of "a" when they are talking about projectiles.
Login to Lecture OnLine and in the 2D motion chapter, solve: Pebble Toss