Fireworks: The Localized Big Bang

Topics

Behaviors of quadratic functions, geometric transformations, standard and general forms of functions

Overview

Commercial fireworks must be carefully timed to ignite as intended. The process involves projectile motion that can be modeled with the quadratic function (ignoring air resistance), as found in the study of dynamics in physics: h = –16t² + v₀t + h₀. We need to study the parameter-behavior connection as each parameter has real-world significance. The contextual situation will lead to the study of behaviors of the quadratic function in standard form too.

The "Hook"

Will I get burned if the person shooting the fireworks does not know about projectile motion? What if the set-up person did not pass Algebra II in high school? What if one of the shells does not explode? How long will it take to hit me? Does the explosion happen on the way up? At the maximum height? Coming down? How long does it take for a fireworks projectile to reach its maximum height?

The Investigation

Typically, we would expect that fireworks are set to explode when the projectile shell carrying the explosives reaches a maximum height, so the view of the explosion is maximized. Secondly, the pyrotechnic operator may want the fireworks to be more of a horizontal (and falling) display – which would also call for an explosion at the maximum point on the trajectory. But, with some types of fireworks, the display looks best if it is moving upwards. The bottom line seems to be that we need to know where the fireworks projectile is at all times and adjust the firing time to the type of fireworks being exploded. For examples of the various types of firework displays, see How Fireworks Work.

The investigation begins with an introduction to the connection between the parameters of a quadratic function and the behavior of the function (see Activity 1). Students then apply these connections to situations involving fireworks (see Activity 2). They conclude the investigation with a detailed exploration of projectile motion using a tossed ball and a motion detector. They use the collected data to develop a mathematical model to describe the motion (see Activity 3).

Teaching Tips

• What basic physics tells us is that the time-height relationship of a projectile being shot straight up is h = –16t² + v₀t + h₀, where h is the height of the projectile at time t; h₀ is the initial height of the projectile; v₀ is the initial velocity of the projectile obtained from the explosion in the tube that holds the projectile; and –16 represents ½ g – the Earth’s gravitational acceleration due to gravity. On other planets, ½ g takes on other values due to differing planet masses. (Note: the model of projectile motion is not exact due to a slight decrease in height from air resistance.)  
• Since we cannot legally shoot fireworks in the classroom, we will create a projectile by tossing a ball straight up. While the ball is in flight, we will collect time-height data with the CBR2, or CBL2 and a motion sensor. From the model h = –16t² + v₀t + h₀, we will need to make conjectures for parameters v₀ and h₀. The initial height should be relatively simple to get, but v₀ will amount to uneducated, yet instructive, guessing. At the same time, if we want to model the actual time-height data, we will find that the parameter –16 may not be exact due to air resistance. That is, the “fluffier” the ball used, the more air resistance will alter the quadratic model. A second option may be to use the model y = d(x + e)² + f for the projectile motion. We know that d is about –16, and the collected data gives us e and f.
• All activities REQUIRE a graphing calculator.  
• Encourage students to look for patterns as they answer questions. There are opportunities to generalize based on patterns established in the questions. The ability to generalize may be more important than finding answers.  
• The first and third student activities are teaching in nature and do not require prerequisites other than the teacher activity. The modeling project is summative in nature and should be assigned at the end of the quadratic function unit. 
• Activities should be assigned to small groups, but individual student use is OK too. Consider using the first and third student activities as a part of homework; that is, give them to students at the end of a class period and collect them at the beginning of the next class session. 
• The modeling project may be assigned with a one-week deadline. Students are expected to work on the project just as working adults may do – with an established deadline and access to resources and experts (maybe parents?) in order to complete the project.
• Solutions to the activity sheets can be found here.

Supporting Resources

For background information on fireworks, see How Fireworks Work.

Citation

Adapted from Foundations for College Mathematics (2nd ed.) & ancillary workbook, 2008, by E. D. Laughbaum.