Definitions
The ballistic trajectory of a projectile is the path that a thrown object will take under the action of gravity, neglecting all other forces, such as friction from air resistance, or propulsion.
In physics, a projectile launched with specific initial conditions in a uniform gravity field will have a predictable range.
Topics of Interest
This article provides a list of methods for calculating the trajectory and range of a projectile under the influence of Earth's gravity.
In the equations on this page, the following variables will be used:
 g: the gravitational acceleration—usually taken to be 9.81 m/s^{2} near the Earth's surface
 θ: the angle at which the projectile is launched
 v: the velocity at which the projectile is launched
 y_{0}: the initial height of the projectile
 d: the total horizontal distance traveled by the projectile
Conditions at the final position of the projectile
Distance traveled
The total horizontal distance (d) traveled.
When the surface the object is launched from and is flying over is flat, the distance traveled is:
As a special case, the distance is given by
when the angle (θ) is 45° and the initial height (y_{0}) is 0.
Time of flight
The time of flight (t) is the time it takes for the projectile to finish its trajectory.
As above, this expression can be reduced to
if θ is 45° and y_{0} is 0.
Angle of reach
The "angle of reach" (not quite a scientific term) is the angle
(θ) at which a projectile must be launched in order to go a distance d, given the initial velocity v.
Range of a Projectile
In physics, a projectile launched with specific initial conditions in a uniform gravity field will have a predictable range. As in Trajectory of a projectile, we will use:
 g: the gravitational acceleration—usually taken to be 9.81 m/s^{2} near the Earth's surface
 θ: the angle at which the projectile is launched
 v: the velocity at which the projectile is launched
 y_{0}: the initial height of the projectile
 d: the total horizontal distance travelled by the projectile
When neglecting air resistance, the range of a projectile will be
If (y_{0}) is taken to be zero, meaning the object is being launched on flat ground, the range of the projectile will then simplify to
For more calculations:
https://en.wikipedia.org/wiki/Range_of_a_projectile
https://en.wikipedia.org/wiki/Trajectory
Source: Wikipedia (All text is available under the terms of the GNU Free Documentation License and Creative Commons AttributionShareAlike License.)
