MOTION OF THE BODY IN GRAVITY FIELD

Let us consider the motion of free body in the presence of the gravitational forces. If the cannon is located at the point with coordinates (0, 0, 0), then the shell fired will move by the trajectory, which can be described by the following equations

X = (v cosj) t
Y = (v sinj) t - gt²/2,

where v is the initial velocity of the shell along the gun tube, j is the angle between the gun tube and the horizon (X-axis), t is time, g is the acceleration of the free fall in the gravitational field of the Earth. Substituting t from the first equation into the second one we can find the expression for the trajectory of the shell:

Y = X tgj - (g/2v²)(1 + tg²j) X²

This means that the trajectory is of parabolic form. We can find from this equation the maximal range of a shot X_max (when Y=0) and the maximal height of the trajectory Y_max (when the first derivative of Y equals to 0):

X_max = v²sin(2j)/g
Y_max = v²sin²j/2g

We can see from the first of these two equations that the maximal range of the shot is achieved when the angle j equals to 45 degrees. The maximal height of the flight is achieved when we shoot vertically. Video-animation shows the cases of the shooting at angles 30, 45, and 70 degrees.