PYROTECHNICS EXAMPLES
1. Independence Day celebrations usually have fireworks displays. Pyrotechnics is
the scientific name for fireworks. This word comes from Greek words meaning,
“fire arts.”
2. Factoring can be used to solve problems dealing with Pyrotechnics. Think about
the following question. A certain fireworks rocket is set off at an average velocity
of 440 feet per second. This type of fireworks is designed to explode at a height of
3000 feet. How many seconds after it is set off will it reach 3000 feet and
explode?” (See # 10 below.)
3. Ask students to analyze the following products.
a. 3(0) = 0
a) 0(-8) = 0
1
1
b) [ + − ](0) = 0
2
2
c) (0)(x + 2) = 0
4. Ask students what they noticed about the products above. Get student responses.
(Help students recognize that all the products equal zero. Notice that in each case
at least one of the factors is zero.)
5. Introduce the students to the “Zero Product Property.”
For all numbers “a” and “b”, if ab = 0,
then a = 0 or b = 0, or both “a” and
“b” equal zero.
6. Introduce the students to formula Îh = vt – 16t2.
If an object is launched from ground level, it reaches its maximum
height in the air at a time halfway between launch and impact
times. Its height above the ground after “t” seconds is give by the
formula “h = vt – 16t2”. In this formula, “h” represents the height
of the object and “v” represents the initial velocity.
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7. Example: A flare is launched from a life raft with an initial velocity of 144 feet
per second. How many seconds will it take for the flare to return to the sea?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “h” represents the height of the flare
in feet when it returns to the sea. Thus, h = 0.
The variable “v” represents the initial velocity of
the flare, in feet per second. Thus, v = 144.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
0 = 144t – 16t2
0 = 16t(9 – t)
16t = 0 or 9 – t = 0
t=0
or 9 = t
EXAMINE Î The flare returns to the sea in 9
seconds. The answer “0” is not a
reasonable answer since it represents
the time when the flare is launched.
Pyrotechnics
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8. Example: A flare is launched from a life raft with an initial velocity of 160 feet
per second. How many seconds will it take for the flare to return to the sea?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “h” represents the height of the flare
in feet when it returns to the sea. Thus, h = 0.
The variable “v” represents the initial velocity of
the flare in feet per second. Thus, v = 160.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
0 = 160t – 16t2
0 = 16t(10 – t)
16t = 0 or 10 – t = 0
t=0
or 10 = t
EXAMINE Î The flare returns to the sea in 10
seconds. The answer “0” is not a
reasonable answer since it represents
the time when the flare is launched.
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9. When will the flare, in # 8 above, be 336 feet off the ground?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “h” represents the height of the flare
in feet when it reaches 336 feet. Thus, h = 336.
The variable “v” represents the initial velocity of
the flare in feet per second. Thus, v = 160.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
336 = 160t – 16t2
16t2 – 160t + 336 = 0
16(t2 – 10t + 21) = 0
16(t – 7)(t – 3) = 0
(t – 7) = 0 or (t – 3) = 0
t=7
or t = 3
EXAMINE Î The flare is at a height of 336 feet 3
seconds (on the way up) after launch
and 7 seconds (on the way down)
after launch.
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10. Example: A certain fireworks rocket is set off at an average velocity of 440 feet
per second. This type of fireworks is designed to explode at a height of 3000 feet.
How many seconds after it is set off will it reach 3000 feet and explode?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “h” represents the height of the
fireworks in feet when it reaches 3000 feet. Thus,
h = 3000.
The variable “v” represents the initial velocity of
the fireworks in feet per second. Thus, v = 440.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
3000 = 440t – 16t2
16t2 – 440t + 3000 = 0
8(2t2 – 55t + 375) = 0
8(2t – 25)(t – 15) = 0
2t – 25 = 0 or t – 15 = 0
2t = 25
or t = 15
t = 12.5
EXAMINE Î The fireworks reaches 3000 feet at 12.5 seconds and 15 seconds.
The answer we must accept is 12.5 seconds since the rocket can
explode only once.
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Name:__________________
Date:____________
Class:__________________
PYROTECHNICS WORKSHEET
Use “h = vt – 16t2” to find the missing quantity. Show steps and check answer to see if it
makes sense.
1. Find “v” if t = 5 seconds and h = 480 feet.
2. Find “h” if t = 7 seconds and v = 1700 feet per second.
3. Find two values for “t” if v = 160 feet per second and h = 336 feet.
Solve each of the problems below. Use the 4-step approach to problem solving:
a.
b.
c.
d.
Explore “Define a variable.”
Plan “Write an equation.”
Solve “Solve the equation and answer the problem.”
Examine “Check to see if the answer makes sense.”
4. At the 4th of July Celebration, a “Red Dragon” streamer was launched with an
initial velocity of 192 feet per second. How many seconds will it take the streamer
to return to the ground? How long did it take the streamer to reach its maximum
height? What was the maximum height?
5. The New Year Celebration started with the firing of a “Lightning Burst” rocket.
The rocket exploded at its maximum height. What was the maximum height, and
how long was it in the air when it exploded if the initial velocity was 64 feet per
second?
6. A rocket is fired with an initial velocity of 1920 feet per second. When will it be
32,000 feet high?
7. A flare is launched with an initial velocity of 128 feet per second. How many
seconds will it take the flare to return to the sea?
Pyrotechnics
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PYROTECHNICS WORKSHEET KEY
Use “h = vt – 16t2” to find the missing quantity. Show steps and check answer to see if it
makes sense.
1. Find “v” if t = 5 seconds and h = 480 feet.
h = vt – 16t2
480 = v(5) – 16(5)2
480 = 5v – 400
880 = 5v
176 = v
Check Îh = vt – 16t2
Is 480 = (176)(5) –
16(5)2
Is 480 = 880 – 400
Is 480 = 480 (YES)
Velocity is 176 feet per second.
2. Find “h” if t = 7 seconds and v = 1700 feet per second.
h = vt – 16t2
h = 1700(7) – 16(7)2
h = 11900 – 784
h = 11,116
Check Îh = vt – 16t2
Is 11116 = (1700)(7) – 16(7)2
Is 11116 = 11900 – 784
Is 11116 = 11116 (YES)
Height is 11,116 feet.
3. Find two values for “t” if v = 160 feet per second and h = 336 feet.
h = vt – 16t2
336 = 160(t) – 16(t)2
16t2 – 160t + 336 = 0
16(t2 – 10t + 21) = 0
16(t – 3)(t – 7) = 0
t – 3 = 0 or t – 7 = 0
t=3
or t = 7
Check Îh = vt – 16t2
Is 336 = (160)(3) – 16(3)2
Is 336 = 480 – 144
Is 336 = 336 (YES)
Is 336 = (160)(7) – 16(7)2
Is 336 = 1120 – 784
Is 336 = 336 (YES)
At 3 seconds and 7 seconds, the rocket
is 336 feet above the ground.
Pyrotechnics
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Solve each of the problems below. Use the 4-step approach to problem solving:
a.
b.
c.
d.
Explore “Define a variable.”
Plan “Write an equation.”
Solve “Solve the equation and answer the problem.”
Examine “Check to see if the answer makes sense.”
4. At the 4th of July Celebration, a “Red Dragon” streamer was launched with an
initial velocity of 192 feet per second. How many seconds will it take the streamer
to return to the ground? How long did it take the streamer to reach its maximum
height? What was the maximum height?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “v” represents the initial velocity of the Red
Dragon. Thus, v = 192.
The height “h” will equal zero “h = 0” when it returns to earth.
If we find “t”, we know that it ascended for one-half of this
time and descended for the other one-half.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
0 = 192t – 16t2
0 = 16t(12 – t)
16t = 0 or 12 – t = 0
t=0
or 12 = t
Since Red Dragon was in the air for 12
seconds, it ascended for 6 seconds and
descended for 6 seconds.
It took 6 seconds to reach maximum height.
SOLVE Î h = vt – 16t2
Disregard t = 0 since that is when Red
Dragon is on the ground.
Red Dragon was in the air for 12
seconds.
h = 192(6) – 16(6)2
h = 1152 – 576
h = 576 feet
EXAMINE Î Is 0 = 192(12) – 16(12)2 ÎIs 0 = 2304 – 2304 (YES)
Is 576 = 192(6) – 16(6)2 ÎIs 576 = 1152 – 576 Î Is 576 = 576 (YES)
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5. The New Year Celebration started with the firing of a “Lightning Burst” rocket.
The rocket exploded at its maximum height. What was the maximum height, and
how long was it in the air when it exploded if the initial velocity was 64 feet per
second?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “v” represents the initial velocity of the Lighting
Burst. Thus, v = 64.
First, find out how long the rocket was in the air. Then use onehalf of this amount for time it ascends.
The height “h” will equal zero “h = 0” when it returns to earth.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
0 = 64t – 16t2
0 = 16t(4 – t)
16t = 0 or 4 – t = 0
SOLVE Î Since the rocket was in the
air for two seconds, we can substitute:
h = vt – 16t2
t=0
h = 64(2) – 16(2)2
h = 128 – 64
h = 64 feet
or 4 = t
Disregard t = 0 since that is the time
when the rocket is on the ground.
Now take one-half of (4) = 2.
The rocket ascended for 2 seconds then
exploded.
The height that the rocket exploded was
64 feet.
EXAMINE Î Is 0 = 64(4) – 16(4)2 ÎIs 0 = 256 – 256 (YES)
Is 64 = 64(2) – 16(2)2 ÎIs 64 = 128 – 64 Î Is 64 = 64 (YES)
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6. A rocket is fired with an initial velocity of 1920 feet per second. When will it be
32,000 feet high?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “v” represents the initial velocity of the rocket.
Thus, v = 1920.
The variable “h” represents the height. Thus, h = 32,000.
time and descended for the other one-half.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
32000 = 1920t – 16t2
16t2 – 1920t + 32000 = 0
16(t2 – 120t + 2000)
16(t – 20)(t – 100)
t – 20 = 0 or t – 100 = 0
t = 20
or
t = 100
The rocket passes the 32,000 feet mark
at 20 seconds and at 100 seconds.
EXAMINE Î h = vt – 16t2
Is 32000 = 1920(20) – 16(20)2
Is 32000 = 38400 – 6400
Is 32000 = 32000 (YES)
Pyrotechnics
EXAMINE Î h = vt – 16t2
Is 32000 = 1920(100) – 16(100)2
Is 32000 = 192000 – 160000
Is 32000 = 32000 (YES)
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7. A flare is launched with an initial velocity of 128 feet per second. How many
seconds will it take the flare to return to the sea?
EXPLORE ÎUse the formula h = vt – 16t2
The variable “h” represents the height of the flare
in feet when it returns to the sea. Thus, h = 0.
The variable “v” represents the initial velocity of
the flare in feet per second. Thus, v = 128.
PLAN ÎSubstitute the appropriate values into the formula “h = vt – 16t2”.
SOLVE Î h = vt – 16t2
0 = 128t – 16t2
0 = 16t(8 – t)
16t = 0 or 8 – t = 0
t=0
or 8 = t
EXAMINE Î The flare returns to the sea in 8
seconds. The answer “0” is not a
reasonable answer since it represents
the time when the flare is launched.
Pyrotechnics
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Student Name: __________________
Date: ______________
PYROTECHNICS CHECKLIST
1. On question 1, did the student find the missing quantity, show steps, and check
answer to see if it made sense?
a. Yes (15 points)
b. Student found missing quantity but did not show steps or did not check to
see if answer made sense (10 points)
c. Student showed steps but missing quantity was incorrect (5 points)
2. On question 2, did the student find the missing quantity, show steps, and check
answer to see if it made sense?
a. Yes (15 points)
b. Student found missing quantity but did not show steps or did not check to
see if answer made sense (10 points)
c. Student showed steps but missing quantity was incorrect (5 points)
3. On question 3, did the student find the missing quantity, show steps, and check
answer to see if it made sense?
a. Yes (15 points)
b. Student found missing quantity but did not show steps or did not check to
see if answer made sense (10 points)
c. Student showed steps but missing quantity was incorrect (5 points)
4. On question 4, did the student solve problem correctly using the 4-step approach
to problem solving?
a. Yes (15 points)
b. Student found correct solution but did not use the 4-step approach to
problem solving (10 points)
c. Student used 4-step approach to problem solving but solution was
incorrect (5 points)
5. On question 5, did the student solve problem correctly using the 4-step approach
to problem solving?
a. Yes (15 points)
b. Student found correct solution but did not use the 4-step approach to
problem solving (10 points)
c. Student used 4-step approach to problem solving but solution was
incorrect (5 points)
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6. On question 6, did the student solve problem correctly using the 4-step approach
to problem solving?
a. Yes (15 points)
b. Student found correct solution but did not use the 4-step approach to
problem solving (10 points)
c. Student used 4-step approach to problem solving but solution was
incorrect (5 points)
7. On question 7, did the student solve problem correctly using the 4-step approach
to problem solving?
a. Yes (15 points)
b. Student found correct solution but did not use the 4-step approach to
problem solving (10 points)
c. Student used 4-step approach to problem solving but solution was
incorrect (5 points)
Total Number of Points _________
A
94 points and above
B
84 points and above
C
74 points and above
D
63 points and above
F
62 points and below
Pyrotechnics
Any score below C
needs
remediation!
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