It is possible to demonstrate that the maximum distance occurs when the angle at which the projectile is fired is 
.
In fact, the laws of motions on both x- and y- directions are
From the second equation, we get the time t at which the projectile hits the ground, by requiring
, and we get:
And inserting this value into Sx(t), we find
And this value is maximum when
, so this is the angle at which the projectile reaches its maximum distance.
So now we can take again the law of motion on the x-axis
And by using
, we find the value of the initial velocity v0:
In fact, the laws of motions on both x- and y- directions are
From the second equation, we get the time t at which the projectile hits the ground, by requiring
And inserting this value into Sx(t), we find
And this value is maximum when
So now we can take again the law of motion on the x-axis
And by using