Parabolas and fireworks
Last semester, I taught a course called “Learning Processes in Math and Science.” Public and private grade school, high school and college teachers came together to discuss how children learned these two subjects, and how to motivate them to do so.
I said one way to encourage students was to help them apply abstract concepts to real life. I asked the teachers to prepare pertinent lesson plans and told them I would feature the most innovative ideas from time to time.
Florante Belardo, who teaches at Miriam College High School, has a lesson plan on using fireworks to introduce parabolas. It is suitable for second-year high school students.
The World Pyro Olympics is an annual pyrotechnics exhibition usually held at SM Mall of Asia. Companies from different countries showcase their products. The finale is a fireworks display to the accompaniment of music.
To synchronize fireworks explosion with the music, a lot of mathematics, physics and experimentation are put to work. We investigate how this is done by using quadratic equations and projectile motion. A projectile is any object thrown into space with force, such as a rocket.
Look at the following two equations.
In algebra, the standard form of a quadratic equation is y = ax2 + bx + c.
In physics, the kinematics formula for a projectile thrown upward is y = 0.5g t2 + v t + h, where g is the acceleration due to gravity (assumed to be -10 meters per second squared), v is the initial velocity of the projectile (the launch speed, in meters per second), y is the height the projectile reaches (in meters), t is time in seconds and h is the initial height of the projectile (in meters).
Do you notice anything similar with the two equations?
The second equation is just a special case of the first, with a = 0.5g, b = v, x = t and c = h. Thus, projectile motion is modeled by a quadratic equation.
During the World Pyro Olympics, safety standards are enforced. Exhibitors must launch their fireworks to a height of 35 meters so people can view them properly and avoid debris from the explosion.
Fireworks are launched from a barge at Manila Bay, with a height of 2 meters from the surface of the water. The shells are launched with an initial velocity of 30 meters per second. Plug in these values into the equation and complete the table below.
0 1 2 3 4 5 6
Height (y) in meters
For example, if t = 2, v = 30, h = 2, then y = 0.5 (-10)(2)2 + (30)(2) + 2. Thus y = 42 meters per second. Do the same for the other values of t.
Will the fireworks go beyond 35 meters?
Plot the points from the table on the plane in graphing paper, and look at the graph of the quadratic function.
What is the shape of the graph? The graph is a parabola.
Looking at the graph, answer the following questions: After how many seconds did the fireworks attain the greatest height? What was the maximum height? Record your answers.
The time it takes for the firework to reach maximum height can also be found with the formula -b/2a. Use the values above and plug into the formula to get time t = -30/2(0.5)(-10) = 3 seconds.
Once you have gotten time t, plug back into the quadratic equation above to solve for y (height). Record your answers.
Did you get the same set of answers?
Afterwards, teachers can continue with the formula for vertex of a parabola, and emphasize that projectiles attain their maximum height at the vertex.
Some types of fireworks explode low, while the larger and more powerful ones must be launched to greater heights. Here is a list of some fireworks and their launch speeds. Assume that they explode when they reach maximum height. Determine the greatest heights they achieve.
Kamuro is a Japanese word meaning “boy’s haircut.” A dense burst of silver or gold stars leaves a trail in the sky, looking like a boy’s haircut when fully exploded. Launch speed is 32 meters per second.
Crossette is a shell with many large stars that travel a short distance before breaking apart into smaller ones, creating grids in the air. Launch speed is 24 meters per second.
Spider contains a fast-burning component to enable stars to travel in a straight line before burning out, resulting in a series of lines much like a spider’s legs. Launch speed is 34 meters per second.
Horsetail, also known as waterfall, has long-burning tailed stars that go a short way from the shell before free-falling, creating the appearance of a horse’s tail. Launch speed is 26 meters per second.
Time rain is caused by big slow-burning stars in a shell that sizzle and leave sparks behind, resulting in beautiful starry rain. Launch speed is 28 meters per second.
Multibreak shell contains several smaller shells of many sizes and kinds. The initial burst scatters them across the sky before exploding. Launch speed is 40 meters per second.
After determining the heights reached, arrange the fireworks based on maximum height attained.
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