1. First we need to find how long the cannonball is in the air, its "hang time" if you will. This span of time is the time it takes for the cannonball to go up, and fall back down. This is completely dependent on the y-component of the velocity, so we have to use some trigonometry to find this. If you drew a right triangle, you'll find that the y-comp onent:
Velocity at y axis = (Initial Velocity)sin(79) = 993sin79 = 974.756 m/s.
2. Now that you have the velocity at the y axis, We can now find the time it takes for the cannon to land. We can use the equation
Final Velocity = Initial Velocity + (Acceleration)(Time)
Vf = Vi + at
note that your aceeleration = -g, since it is pulling it downward so:
Vf = Vi - gt.
Vf + gt = Vi
gt = Vi - Vf
t = (Vi - Vf)/g
We know the Initial speed in the y axis, which is 974.756 m/s. But what about the final velocity? Here's something interesting: When you throw a ball up at say a certain speed 2m/s, when it falls back to your hand (assuming you kept your hand at the same place) it will return to your hand at a speed of -2m/s, the opposite of the initial velocity. So in the case of the cannonball, I can say that the final velocity is -974.756 m/s. I know my Vi, Vf, and g, so I can now solve for time.
t = (974.756 - (-974.756))/9.8 = (974.756 + 974.756)/9.8 = 1949.512/9.8 = 198.9298 seconds.
3. Why do I need to know the hang time? This is because the ball is traveling at the x direction for this much time, and after that time, it stops moving (unless it keeps on rolling, but that's another problem! Let's assume it stays put where it lands.)
Anyway, I can use the equation
speed = distance/time
distance = (speed)(time) = vt
I have my time, but I need my velocity. Remember we're looking for the distance at the x-direction, not the y, so we for the speed here, we need to get the x-component of the projectile's velocity. Using trigonometry again:
Velocity at x axis = (Initial Velocity)cos(79) = 993cos79 = 993(0.19) = 189.473 m/s.
So, using this on the previous equation, and the air-time:
distance traveled along the horizontal direction = vt = (189.473)(198.9298) = 37,691.826 meters.