In simple terms, there are three initial conditions to consider if you’re going to fire a cannon: the weight of the cannonball, how much gunpowder you load into the cannon, and the angle of the cannon when you fire it.
If you’re trying to figure out what effect changing those variables can have, the right way to do it is to fix two of them and then see what happens as you vary the third.
For example, set the cannon at a 30-degree angle, and use the same weight of cannonball for 5 shots. Pack each of those 5 shots with increasing amounts of gunpowder… like 10, 20, 30 pounds and so on.
After you’ve fired those five cannonballs, measure the different distances and graph them. And draw a line through those 5 points… and extend it, beyond the last one, following the shape of that line. It might be perfectly straight. It might curve a bit. This is called extrapolation, and lets you make a pretty good guess as to what would happen if you had kept adding more gunpowder.
Now, do the same… this time, use the same amount of gunpowder, but use different weights of cannonballs. Graph and extrapolate that too.
Finally, pick one of those cannonball weights and a fixed amount of gunpowder, and fire them all, changing the cannon’s angle by 5 degrees each time. Graph and extrapolate.
Given those three graphs and their extrapolated lines, you now have a pretty good idea of how to fire this cannon, depending on what you desire. There may be many ways of hitting a target 500 yards away, but one uses more gunpowder. Or maybe you want to hit it with a bigger cannonball. Maybe there are trees in the way, so you’ll need a steeper angle.
One thing that’s certain; the only control you have with this cannonball is what you set with these initial conditions. Once you light that fuse and the cannoball blasts its way out of there, there is nothing you can do about its trajectory. Hopefully you got it right.
It occurs to me that a more modern and relevant example would be golf. When you’re trying to hit a golfball into a hole 150 yards away, there are many variables to consider, and usually, too many for most … [Continue Reading]