Physics of pyrotechnics
It is known what every type of shell has certain initial speed, it'll be used in calculations. These velocities are the speeds that the shells are traveling as they are fired out of the mortar. The 2" through 6" shells are used about at almost all fireworks shows and are used almost exclusively at small shows. The 8", 10", and 12" shell sizes are usually used at only large fireworks shows as they are more expensive than the smaller sizes.
The 24" and 36" shell sizes are even more expensive because they produce extremely large burst patterns. These monstrous aerial shells are only used at the largest shows and during special circumstances. As you can see in the table, larger shell sizes produce greater initial mortar velocities. This happens because the larger mortars used to fire larger shells have the capacity to house greater amounts of blackpowder/pyrex used to propel the shells out of the mortar.
Greater amounts of blackpowder/pyrex, when burned, produce more excess gases than do smaller amounts. These larger amounts of excess gases cause the shell to be pushed or propelled out of the mortar faster, resulting in greater initial velocities. The greater initial velocities produced by larger shells result in the shell attaining a greater height before it explodes and emits its bright flash of light. Shells usually travel about 100 feet vertically for every inch they are in diameter; depending on the angle they are fired from.
The relationships between the initial velocities and the distances traveled by the shells can be understood and manipulated by using the following formulas and mathematical methods:
Y=VyT+0.5GT2
Y = vertical height, Vy = initial vertical velocity, T = hang time, G = acceleration due to gravity
X=VxT
X = horizontal distance, Vx = initial horizontal velocity, T = hang time
The Pythagorean Theorem:
a2 + b2 = c2
a or b = vertical or horizontal velocity, c = resultant initial velocity
The Trigonometric Functions - sine, cosine, and tangent
In a right triangle sine = opposite side/hypotenuse, cosine = adjacent side/hypotenuse, tangent = opposite side/adjacent side
The first two formulas you see are primarily used to chart trajectories
like in the graph on the left that shows the flight paths of 2"
through 12" shells fired at 75 degrees. These graphs are very useful
tools that allow pyrotechnicians to visualize how high and how far
their shells will travel during a show. This information can be
used to aid the process of choreographing the show to music, and
determining if some shells will exceed the safe zone for that particular
site.
The Pythagorean Theorem is used to find a certain initial velocity
value if the other two are known. This is helpful in determining
information needed for the other formulas. The Trigonometric Functions
are also used to find initial velocity values, but are used to find
vertical heights, horizontal distances, and firing angles as well.
Pyrotechnicians use these mathematical methods along with charts,
graphs, and computer programs derived from them to plan their impressive
displays.
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