When a projectile moves near the surface of the
earth, the acceleration can be taken as constant, directed down, and equal
to the free fall accelearation of any object. (We ignore wind resistance.)
The free fall acceleration is taken to have magnitude, g. The direction of
the acceleration is down; the negative y direction for the coordinate
system shown in the figure.

Taking x to be horizontal and y to be vertical, the equations for position
components are shown in the figure. Taking the origin to be at the initial
point of the trajectory, we can easily eliminate time from the two
position equations. The result is an equation for the **trajectory; **the
equation that gives the vertical displacment (Y) as a function of
horizontal displacement (X).

We can think of this equation as having the x and y components of initial
velocity (V sub 0) as parameters. Alternatively, we can write the
parameters in terms of the initial velocity magnitude and direction. Both
parameterizations are shown in the figure.