Discussion
introduction
A projectile is any object that is cast, fired, flung, heaved, hurled, pitched, tossed, or thrown. (This is an informal definition.) The path of a projectile is called its trajectory. Some examples of projectiles include…
 a baseball that has been pitched, batted, or thrown
 a bullet the instant it exits the barrel of a gun or rifle
 a bus driven off an uncompleted bridge
 a moving airplane in the air with its engines and wings disabled
 a runner in mid stride (since they momentarily lose contact with the ground)
 the space shuttle or any other spacecraft after main engine cut off (MECO)
The force of primary importance acting on a projectile is gravity. This is not to say that other forces do not exist, just that their effect is minimal in comparison. A tossed heliumfilled balloon is not normally considered a projectile as the drag and buoyant forces on it are as significant as the weight. Heliumfilled balloons can't be thrown long distances and don't normally fall. In contrast, a crashing airplane would be considered a projectile. Even though the drag and buoyant forces acting on it are much greater in absolute terms than they are on the balloon, gravity is what really drives a crashing airplane. The normal amounts of drag and buoyancy just aren't large enough to save the passengers on a doomed flight from an unfortunate end. A projectile is any object with an initial nonzero, horizontal velocity whose acceleration is due to gravity alone.
An essential characteristic of a projectile is that its future has already been preordained. Batters may apply "body English" after hitting a long ball, but they do so strictly for psychological reasons. No amount of leaning to one side will make a foul ball turn fair. Of course, the pilot of a disabled airplane may regain control before crashing and avert disaster, but then the airplane wouldn't be a projectile anymore. An object ceases to be a projectile once any real effect is made to change its trajectory. The trajectory of a projectile is thus entirely determined the moment it satisfies the definition of a projectile.
The only relevant quantities that might vary from projectile to projectile then are initial velocity and initial position
This is where we run into some linguistic complications. Airplanes, guided missiles, and rocketpropelled spacecraft are sometimes also said to follow a trajectory. Since these devices are acted upon by the lift of wings and the thrust of engines in addition to the force of gravity, they are not really projectiles. To get around this dilemma, it is common to use the term ballistic trajectory when dealing with projectiles. The word ballistic has its origins in the Greek word βαλλω (vallo), to throw, and surfaces repeatedly in the technical jargon of weaponry from ancient to modern times. For example…
 The ballista, which looks something like a giant crossbow, was a siege engine used in medieval times to hurl large stones, flaming bundles, infected animal carcasses, and severed human heads into fortifications. Before the invention of gunpowder, ballistas (and catapults and trêbuchets) were the weapons of choice for conquerors.
 An intercontinental ballistic missile is a device for delivering nuclear warheads over long distances. At the start of its journey an ICBM is guided by a rocket engine and stabilizer fins, but soon thereafter it enters the phase of its journey where it is effectively in free fall, traveling fast enough to keep it above the earth's atmosphere for awhile but not fast enough to enter orbit permanently. The adjective "intercontinental" refers to the long range capabilities, while the largely free fall journey it takes makes it "ballistic". ICBMs are the ultimate killing machines, but they have never been used in combat to date.
The wide geographic range as well as the wide historic range of these things we call projectiles raises some problems for the typical student of physics. When a projectile is sent on a very long journey, as is the case with ICBMs, the magnitude and direction of the acceleration due to gravity changes. Gravity isn't constant to begin with, but the effect is not very pronounced over everyday ranges in altitude. From the deepest mines in South Africa to the highest altitudes traversed by commercial airplanes, the magnitude of the acceleration due to gravity is always effectively 9.8 m/s^{2} ± 0.05 m/s^{2}. Similarly, unless you routinely travel medium to long distances, you aren't likely to experience much of a change in the direction of gravity either. To experience a 1° shift in "down" would require traveling ^{1}/_{360} of the circumference of the earth — roughly 110 km (70 mi) or the length of a typical morning commute to work in Southern California. Thus for projectiles that won't rise higher than an airplane nor travel farther than the diameter of L.A., gravity is effectively constant. This covers the first five of the examples described at the beginning of this section (baseballs, bullets, buses in actionadventure movies, distressed airplanes, and joggers) but not the sixth (the space shuttle after MECO).
To distinguish such simple projectiles from those where variations in gravity and the curvature of the earth are significant, I propose using the term simple projectile. For the remaining problems, the term general projectile seems appropriate since a general solution in mathematics is one that also includes the special cases, but I'm less adamant about this term.
Consider an effectively spherical earth with a single tall mountain sticking out of it like a giant tumor. Now imagine using this location as a place to launch projectiles horizontally with varying initial velocities. What effect would velocity have on range? Well obviously fast projectiles will travel farther than slow ones. A basic concept associated with speed is that "faster means farther", but the relationship is only approximately linear on a spherical earth. For awhile, doubling speed would mean doubling distance, but eventually the curvature of the earth would start to mess things up. At some speed our hypothetical projectile would make it a quarter of the way around the earth and then half way around and then eventually all the way around. At this point our general projectile ceases to be an object with a launch point and a landing point and it starts being a satellite, permanently circling the earth, perpetually changing direction and thus accelerating under the influence of gravity, but never landing anywhere. Technically, such an object would still be a general projectile, since gravity is the primary source of its acceleration, but somehow this doesn't seem right. Objects traveling through what we call "outer space" hardly seem like projectiles any more. They seem like they reside more in the realm of celestial mechanics than terrestrial mechanics. Such distinctions are arbitrary, however, as there is only one mechanics. The laws of physics are assumed universal until it can be demonstrated otherwise. The unification of physical law is a theme that surfaces from time to time in physics.
A projectile and a satellite are both governed by the same physical principles even though they have different names. A simple projectile is made mathematically simple by an idealization (basically a lie of convenience). By assuming a constant value for the acceleration due to gravity, we make the problem easier to solve and (in many cases) do not really lose all that much in the way of accuracy.
Every projectile problem is essentially two onedimensional motion problems…
The kinematic equations for a simple projectile are those of an object traveling with constant horizontal velocity and constant vertical acceleration.
equation  horizontal  vertical  

acceleration  a_{x}  = 0  a_{y}  = −g 
velocitytime  v_{x}  = v_{0x}  v_{y}  = v_{0y} − gt 
displacementtime  x  = x_{0} + v_{0x}t  y  = y_{0} + v_{0y}t − ½gt^{2} 
velocitydisplacement  v_{y}^{2}  = v_{0y}^{2} − 2g(y − y_{0}) 
finish …
The trajectory of a simple projectile is a parabola.
calculus, but not really
max range at 45°, equal ranges for launch angles that exceed and fall short of 45° by equal amounts (ex. 40° & 50°, 30° & 60°, 0° & 90°)
Use the horizontal direction to determine the range as a function of time…
x =  x_{0} + v_{0x}t + ½a_{x}t^{2}  
x =  0 + (v cos θ) t + 0  
x_{final} =  (v cos θ) t_{final} 
Use the vertical direction to determine the time in the air…
y  = y_{0} + v_{0y}t + ½a_{y}t^{2}  
y  = y_{0} + (v sin θ)t − ½gt^{2}  
0  = 0 + (v sin θ)t_{final} − ½gt^{2}_{final}  
t_{final} = 

Combine these two equations…
x_{final} =  (v cos θ)  2(v sin θ)  
g  
x_{final} =  v^{2} sin 2θ  
g  
x_{max} = 
