Photo courtesy of Kevin Dooley

Here in the states, today is Independence Day — the 236 anniversary of the signing of the Declaration of Independence. (Yes, I subtracted 1776 from 2012.) Most of us are taking the day off, but there is one industry that is working overtime: the guys and gals who choreograph and conduct fireworks displays. These gorgeous displays are patriotic, fantastical and downright dangerous. As you can imagine, there’s a ton of math that goes on to make sure that no one in the 500,000-person crowd at the National Mall in Washington, D.C. aren’t injured by the 66,000 pounds of explosives that go off in a 20-minute show. (Oh, and for any of you math teachers out there reading, this is how to get pyrotechnic teens interested in algebra. When they ask when they would ever use conic sections or quadratic equations, talk to them about fireworks and punkin chunkin.) So we’re not going to get into the nitty gritty of the math here. Instead, let’s look at the concepts behind the math involved. First, you need to know how fireworks are set off.  The shell is set in a mortar tube, which rests on the ground. When the fuse is lit, a chemical reaction forces the shell into the air, following a predictable path. As long as everything is timed and spaced properly, the shell bursts and the debris begins to fall back to the ground. You can replicate this (safely) with a tennis ball. Throw it up in the air and watch what happens. You’ll notice that the ball rises and, once it hits a certain height, starts to fall again. If you throw it straight up, it will go higher. If you throw it at an angle, it goes farther out. (Parents: This is a really cool experiment for kids. Have them try throwing the ball at a number of different angles. What happens? Estimate the angle at which you’re throwing the ball. (Straight up and down is 90 degrees.) Then measure the distance from where you threw the ball and where it landed. What kinds of connections can you make between the two?) This is called a trajectory. Physics dictates that the path an object takes when launched into the air will be a curve. Specifically, this curve is a parabola.

The water in this water fountain forms a parabola. (Photo courtesy of Paul Anderson)

Here’s the math part: Every curve has an equation associated with it. That equation describes all sorts of things — like how tall and wide the curve is. But why do fireworks geeks care? Because the equation keeps everyone safe. The firework must be launched at the correct angle, or it could land in the middle of the watching crowd. This magic number depends on the firework in question. Heavier explosives must have greater force behind them. They need that velocity to get them to the right height. Second math part: These equations are always quadratic. In other words, their highest exponent is 2, like this:

x2 + 3x – 9 = 0

For most of the population, solving this equation isn’t important. But I do think it can be useful to know a few things:

1.  Linear equations don’t have exponents,
2.  Curves have exponents, and