I admit that fireworks are not something I think about when the month of July hits only because the sky is so light, the local city waits till January to have all of its fireworks when its much darker. I would like to thank Josh Fisher of the Mathematics Education (K-12) google plus group for this topic.
He provided a link to an article in the Columbus Dispatch which gave a general look at the math that is used in creating fireworks shows. I read the article and I had not realized how much math is involved. Look at this quick list of math they have to use:
1. Trajectories with both a vertical and horizontal element for their motion. This is needed for the fall out zone.
2. Size of the shell - the most common is a 6 to 8 inch shell because the larger the shell, the lift is needed at launch and the larger the explosion.
3. The launch angle which determines how far away the fireworks will explode in relation to the audience, buildings, etc.
4. Air resistance - keeps the shells from traveling too far.
5. A single shell can launch three to five different fireworks that explode at different heights.
6. Humidity or wind can alter the height a shell explodes at.
Think of it this way, a 20 minute show can use 66,000 pounds of explosives. that means they are shooting off 3,300 pounds per minute. So what about the actual equations? Well Inquirer net has a great article on parabolas and fireworks that includes the specific equations.
This particular article begins by comparing the standard quadratic equation with the kinematics formula used for projectiles. The article takes time to define each variable and explaining the values. It has students use the equation to create a graph so students discover for themselves that it is is in a parabolic shape. Furthermore, the activity lists a variety of fireworks with their launch speeds so they can calculate the maximum height of each rocket.