Final answer:
For the cannonball to stay in the air the longest, it should be fired straight up at 90 degrees, but for maximum range without air resistance, the ideal angle is 45 degrees. Air resistance lowers this to approximately 38 degrees. Practical applications like archery or punting in football may require different angles for effectiveness.
Step-by-step explanation:
The question pertains to the ideal angle at which a cannonball should be fired to stay in the air for the longest possible time. The angle that achieves the longest airtime for a cannonball, under ideal conditions that disregard air resistance, is 90 degrees above the horizontal. This is because, at this angle, all of the initial velocity contributes to the vertical component, thus maximizing the time the cannonball spends ascending and descending. However, it is important to note that in the real world with air resistance, this angle could be different. For a cannon firing at a fixed initial speed and ignoring air resistance, the maximum range is achieved when the firing angle is 45 degrees. This optimal range angle changes to approximately 38 degrees when taking air resistance into account. For any given angle not equal to 45 degrees, two angles can result in the same range, and their sum equals 90 degrees.
When considering the practical application of projectile motion, an archer or a football punter might choose different angles based on factors such as wind, obstacles, and the need for height to maximize the range or to decelerate the projectile for a controlled landing.