Projectile motion refers to the motion of an object that is confined
to a plane. In the present case, the body experiences acceleration in one
direction only, the (downward) acceleration due to gravity. The upward
direction is usually chosen to be the y-direction,
while the direction of forward motion of the body is taken to be the x-direction.
The kinematic equations which describe this motion are:
Recall that displacement, velocity and acceleration are vector quantities;
their direction must be accounted for in the above equations. The equations
for the y-direction have been written so that
up is considered the positive direction, hence the negative sign in front
of the acceleration term (gravity acts downward).
If the projectile is fired from a horizontal position, then the y-equations become simplified as all of the initial velocity is in the horizontal or x-direction. By measuring the height from which the projectile was fired, equation (2) can be used to solve for the time of flight of the projectile in terms of the measured height. When placed into equation (1) for t, and combined with the measured distance in the x-direction (the range), the initial y-velocity of the projectile can be determined.
When the progectile is fired at an angle 
, measured from the horizontal, there are components of the initial velocity,
, in both x- and y-directions.
Then Equation (1) is modifed to become:
while equation (2) is modified to look like:
Since the angle and the value of the initial velocity are known, the unknown variables of interest are the time of flight, T, and the range, R. The time of flight can be found from the following argument: If we assume that the projectile returns to the same level from which it starts, then its initial and final y-positions may be taken to be zero. Thus, we get
Factoring out a common t gives:
This has the solution that either t=0 or
We identify the first solution as the initial time of firing, which means that the second solutuon must give the time of flight, i.e.,
When T replaces t in equation (5), we obtain the range,
Using the trigonometric relation
allows for the reduction of equation (11) to
This equation expresses the range as a function of the angle of inclination,
for an inital velocity 
.
