FIREWORKS
Standard form for a quadratic equation is
The parabolic path of a typical fireworks shell is described by the quadratic function y = -16x2 + bx + c.
a = -16 which is the acceleration due to gravity
b is the launch speed in feet per second
c is the height in feet from which the fireworks shell is launched
x is the time in seconds
y is the height (in feet) of the fireworks shell.
During the SummerFest Festivities in Milwaukee, Wisconsin, firework displays are a nightly occurrence on the shore of Lake Michigan. If a firework is launched at a height of 4 feet above the ground with a launch speed of 74 feet per second, the equation represents the firework’s height (y) in feet after (x) seconds. The firework will explode at approximately the highest point. Fire officials require the fireworks to explode at a height greater than 76 feet in order for debris to land in Lake Michigan and not on the festival site. The event planners assured the fire officials that the fireworks would reach a height greater than 76 feet. Complete the table below by substituting the given values into the equation to see if the event planners are correct with their assumption.
time(x) / 0 / 1 / 2 / 3 / 4 / 5 / 6
height
(y)
Are the event planners correct? ______
Plot the points from the table and construct the graph of this quadratic function.
This U-shaped graph is called a parabola.
Looking at the graph, after how many seconds was the greatest height attained by this firework? What was the maximum height?
time = ______height = ______
The answers you found for the time and height in the question above make up the coordinates for the vertex of the parabola. The number of seconds represents the x-coordinate while the height represents the y-coordinate. However, “eyeballing” the graph doesn’t always give an exact location of the vertex.
The vertex can also be found algebraically.
The formula x = will give you the x-coordinate of the vertex. The result of substituting this value back in for x yields the y-coordinate of the vertex.
Use the equation for the vertex to see if it matches the numbers you found above for the time and height. Show your work here.
Listed below are some common fireworks, along with a brief description and the quadratic equation that describes the parabolic path of that particular firework shell if it was launched 4 feet above the ground.
Peony - A spherical break of colored stars that burn without a tail effect. The peony is the most commonly seen shell type.
Chrysanthemum - A spherical break of colored stars, similar to a peony, but with stars that leave a visible trail of sparks.
Dahlia - Essentially the same as a peony shell, but with fewer and larger stars. These stars travel a longer-than-usual distance from the shell break before burning out.
Willow - Similar to a chrysanthemum, but with long-burning silver or gold stars that produce a soft, dome-shaped weeping willow-like effect.
Palm - A shell containing a relatively few large comet stars arranged in such a way as to burst with large arms or tendrils, producing a palm tree-like effect. Proper palm shells feature a thick rising tail that displays as the shell ascends, thereby simulating the tree trunk to further enhance the palm tree effect. You might also see a burst of color inside the palm burst to simulate coconuts.
Ring - A shell with stars specially arranged so as to create a ring. Variations include smiley faces, hearts, and clovers.
Diadem - A type of peony or chrysanthemum with a center cluster of non-moving stars, normally of a contrasting color or effect.
Kamuro - Kamuro is a Japanese word meaning “boys haircut” which is what this shell looks like when fully exploded in the air. A dense burst of glittering silver or gold stars which leave a heavy glitter trail and are very shiny in the night’s sky.
Crossette - A shell containing several large stars that travel a short distance before breaking apart into smaller stars, creating a crisscrossing grid-like effect.
Spider - A shell containing a fast burning tailed or charcoal star that is burst very hard so that the stars travel in a straight and flat trajectory before slightly falling and burning out. This appears in the sky as a series of radial lines much like the legs of a spider.
Horsetail - Named for the shape of its break, this shell features heavy long-burning tailed stars that only travel a short distance from the shell burst before free-falling to the ground. Also known as a waterfall shell.
Time Rain - An effect created by large, slow-burning stars within a shell that leave a trail of large glittering sparks behind and make a sizzling noise. The “time” refers to the fact that these stars burn away gradually, as opposed to the standard brocade “rain” effect where a large amount of glitter material is released at once.
Multi-break Shells - A large shell containing several smaller shells of various sizes and types. The initial burst scatters the shells across the sky before they explode. Also called a bouquet shell.
Fish - Inserts that propel themselves rapidly away from the shell burst, often looking like fish swimming away.
Salute - A shell intended to produce a loud bang rather than a visual effect. Salute shells usually contain flash powder, producing a quick flash followed by a very loud bang. Titanium may be added to the flash powder mix to produce a cloud of bright sparks around the flash. Salutes are commonly used in large quantities during finales to create intense noise and brightness.
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List the fireworks in order (greatest to least) by the maximum height attained.
Firework / Seconds to Explosion [x] / Maximum Height (ft) [y]Which fireworks meet the criteria set forth by fire officials during SummerFest in Milwaukee?
Using the 15 fireworks listed, what is the average amount of time that a fireworks shell is in the air before it explodes?
Make a table of values to graph the parabola of any one of the 15 fireworks.
Name of firework ______
How long does this particular fireworks shell stay in the air? ______
Imagine that you are now part of the event planning staff for SummerFest. The CEO of one of the major sponsors is a big fan of the “horsetail” firework. He demands that the “horsetail” be in the fireworks display this summer or he will discontinue his sponsorship. What ideas could you offer in order to include the “horsetail” in the fireworks display?
BRIEF HISTORY OF FIREWORKS
The chemistry of fireworks has remained basically the same since their discovery. Gunpowder, which is the basis of many fireworks, is supplemented with metal powders to create sparky effects. The three basic ingredients are potassium nitrate, charcoal, and sulfur. Saltpeter enhances the flame. A propellant is used to launch the fireworks into the sky.
The Chinese are credited with discovering fireworks by creating a mixture of charcoal, sulfur, and saltpeter that would explode if enclosed in a small place. The Chinese would use firecrackers to scare away evil spirits. In medieval times, when people spent most of their time in the dark, people were fascinated by the sparks and explosions that created light. During the Renaissance, the military was responsible for both war artillery and for fireworks used during peacetime. In 1575, the Earl of Leicester hosted Elizabeth I and capped a multi-day feast with fireworks. Rulers would often use fireworks in ceremonies to prove they could create magic for their subjects. In the 1730s, fireworks went from being used for dramatic purposes to more common uses.
It wasn’t until the 1830s that pyrotechnicians learned how to add color to fireworks. Replacing potassium nitrate with potassium chlorate, an energetic oxidation agent, raised the combustion temperature of the fireworks to 2000 degrees C (3632 degrees F), allowing for a wider range of colors to be used. Colors are created by adding metal salts to the mixture. Each metal salt produces light in a specific wavelength. Sodium salts create yellow; copper salts create blue; strontium nitrate produces red; barium nitrate makes green; charcoal or other forms of carbon make orange. Modern-day uses range from small to large celebrations. State laws vary concerning the type of fireworks allowed; some states allow none.