Cannonballs may be a thing of human warfare past, but it is not difficult to imagine the sense of dread at seeing a 40-pound metal projectile headed toward your ship, unable to tell for a few agonizing moments whether it will find a deadly mark.
In addition to being the stuff of pirate legends and a prominent feature in video games, including some of the earliest home games of the 1980s, cannonballs provide a superb lesson in the physics of basic projectile motion.
Instead of gold doubloons, though, your "treasure" will come in the form of improved skills at unraveling basic ballistics problems.
History of Cannons and Cannonballs
You won't find cannons or cannonballs for sale at your local big-box retailer these days, but there was a time when having a good cannon on hand was important for a group of individuals possessing, or bent on possessing, a great deal of valuable property. Cannons were once very effective at deterring thieves, or alternatively, making it easier for them to ply their questionable trade.
As you might expect, there is really no such thing as a "typical" cannonball weight. To get a general idea, the British forces of old made cannonballs in discrete masses ranging from about 4 pounds to about 42 pounds. As these were made of iron, a typical density was about 7,860 kg/m3. That's about eight times denser than water.
- Did pirate cannon balls explode? No. They were simply large, heavy, deadly pieces of metal. The vision of exploding cannonballs surely infuses thoughts of centuries-old conflicts on the high seas with drama and intrigue, but this idea fails to accurately reflect reality.
Newton's Equations of Motion
The 17th-century genius Isaac Newton did a lot of things, including inventing many of the methods of modern calculus. But perhaps his greatest feat was deriving the mathematical equations that govern the behavior of objects subjected to the influence of gravity and other forces.
The general solution for the horizontal position x, velocity v and acceleration a of an object at time t is:
This can be combined with other equations of motion to derive related expressions.
The Horizontal Launch
Note that when you fire a cannonball into the distance at a given angle, it is subject to the influence of gravity in exactly the same way it would be if you merely dropped it off the top of a building. That is, if you fire the cannonball with enough force to get it 100 feet into the air, it will fall to the ground just as rapidly if it is moving horizontally while falling as it will if only falling.
One crucial quantity is the range, or the maximum distance the cannonball will travel. This is a function of initial velocity v0 and launch angle θ:
- Because the maximum value of sine is 1, and this occurs at 90 degrees, you can see that the ideal launch angle in terms of maximizing horizontal distance is 45 degrees.
Virtual Cannonballs Game
Toying around with an online cannonball-launching game will not, by itself, make you a physics or ballistics expert. This will, however, give you a chance to gain a sense of how the magnitude and direction of changes to the launch angle and velocity can affect the ultimate path of a projectile, and how early warriors also learned to account for their own mistakes in the course of their bellicose "experiments."
See the Resources for an example of a simple projectile-launching game that allows you to input the launch parameters precisely and observe what happens when you fire another cannonball downrange.