


How Linear are Submarine Missiles?
Developed by Tisha Jones, South Central High School, Winterville, North Carolina
2011 Naval Historical Foundation STEM teacher fellowship



Instructional Goal
In this unit of study students will relate the dimensions and characteristics of submarine ballistic missiles to scatter plots, volume calculations, and linear regression analysis. Students begin the unit by discussing background information on U.S. Navy Fleet Ballistic Missiles and then completing a group activity to build a life size missile model to understand the immense size of these missiles carried in U.S. Navy ballistic missile submarines. Students then read and evaluate a missile data scatterplot. Next, students calculate missile volumes from given missile dimensions. Finally, students use their data to compare missile volume to the distance a missile travels, determining the relationship between volume and distance.
Background on Submarine Missiles
U.S. Navy ballistic missile submarines also called "boomers" carried various types of Fleet Ballistic Missiles (FBM's) during the Cold War (see figure below). The missiles evolved from the single warhead Polaris A1 and A2 to the multiple warhead Polaris A3 through the independently targetable multiple warhead Poseidon A3 and Trident I C4 to the current Trident II D5 missile. The progression from a single warhead to multiple warheads to independently targetable warheads, described below and shown in the three accompanying videos, provided increased target coverage and required larger missiles.
Polaris A1, named for Polaris, the North Star, was a 2stage ballistic missile, powered by solid fuel rocket motors. The first successful underwater launching of a Polaris A1 was conducted in 1960. A video of that launch is included in "All Hands TV: SSBN History", 2:13  2:45. The one minute video: Polaris Launch: "Our Crucial Deterrent" shows the only test launch with a nuclear detonation at the target point. The Polaris A1 was officially retired from active duty in 1965. It had a 1380 mile range.
The Polaris A2 was similar to the A1 except it was longer, allowing it to carry more fuel which made it heavier and increasing the range to 1730 miles. The A2 was first launched in 1961 and was officially retired in 1974. The first Polaris A2 launch is included in "All Hands TV: SSBN History", 2:46  3:22.
The Polaris A3 missile with multiple warheads represented a significantly greater technological advancement over A2 and A1. The first successful launch of the A3 was in 1963. The first Polaris A3 launch is included in "All Hands TV: SSBN History", 3:23  3:46. Its longer range of 2880 miles meant that no land target was beyond reach. The A3 missile was retired in 1982.
The Poseidon C3, named after the mythological Greek god of the sea, had its roots in Polaris technology. It increased in size again, with a 2foot longer length and 20inch greater diameter. Its multiple warheads, each of which could be targeted separately, improved the effectiveness of the FBM weapons system as a deterrent to the outbreak of a nuclear war. The C3s first successful launching was conducted in 1970. The first Poseidon launch is included in "All Hands TV: SSBN History", 3:47  4:32. A twominute video: Poseidon Launch: "Our Crucial Deterrent" chronicles the Cold War period and Poseidon C3 missile launch. The C3 missile was finally retired in 1992, after the Cold War's end.
The Trident I C4 missile name stems from roman mythology. The Trident I C4 missile is a 3stage, solid propellant missile. Its advanced propellant provides a 4600 mile range. The Trident 1 C4 missile was test launched in 1977 and was retired in 2005. The C4 missile's first launch is included in "All Hands TV: SSBN History", 4:33  5:12.
The Trident II D5 missile is also a 3stage, solid propellant missile. The Trident II is more sophisticated and larger than its predecessor Trident I: 10 feet longer and 9 inches greater in diameter. The Trident II D5 was first launched in 1989 near the end of the Cold War. The first D5 launch is included in "All Hands TV: SSBN History", 5:13  6:08. D5 is the only fleet ballistic missile loaded aboard our 14 Tridentclass ballistic missile submarines today.
The purpose of nuclearwarhead tipped fleet ballistic missiles is to deter a nuclear war, and respond if necessary.
Teacher Help 
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Video Resources
All Hands TV: SSBN History: This 2009 All Hands TV looks at the history of the fleet ballistic submarine (SSBN) nuclear deterrence mission. SSBNs have been providing the US with security for many, many years.
Polaris Launch: "Our Crucial Deterrent" (1973): The launch of a Polaris A1 submarine launched ballistic missile from the nuclear submarine USS ETHAN ALLEN, and the ensuing nuclear detonation of its warhead, are shown in this clip from the 1973 U.S. Navy film "Our Crucial Deterrent". Source: Naval History and Heritage Command, Photographic Section, UM23.
Poseidon Launch: "Our Crucial Deterrent" (1973): The 1972 launch of a Poseidon submarine launched ballistic missile is shown in this clip from the 1973 U.S. Navy film "Our Crucial Deterrent". Source: Naval History and Heritage Command, Photographic Section, UM23.

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Blow Up a Missile
Objectives:
 Create scale model "cylinders" the size of a D5 missile, using given dimensions
 Use of simple measuring devices (yard sticks, rulers, measuring tapes)
Materials:
Although the class is working on the same project, it will be split into about 5 groups. Each group is given the tasks to make one part of the missile. Each group will consist of 56 students. As a class you will need the following materials:
 1 roll clear painter's plastic (12X400 is plenty)
 Several pairs of scissors
 5 rolls clear packing tape
 Yard Sticks or Measuring Tapes
 Box Fans
Procedure:
 Discuss as a class the information needed to find the dimensions of the cylinder to build the missile. Discuss the shape as being a cylinder and the 2 dimensional shapes that make a cylinder (rectangle and 2 circles). Using the dimensions: diameter 84 inches and length 44 feet, discuss the amount of painters plastic needed to make each part of the cylinder.
 Instruct groups to cut out the pieces necessary to build specific rectangles and circles. Students will cut out each piece and tape it together so that there are no holes (an airtight bubble except the part to insert the box fans). As they are constructing the cylinder, instruct them to write in permanent marker the dimensions of each piece of plastic.
 Once students have built their cylinder/missile, find a large area to inflate the missile using the fans (example the gym). Stretch the missile across the floor, insert the box fans and tape the plastic to the box fan so that it is airtight. Turn the fan on low speed at first then it may be increased as the bubble starts to inflate more. Hold the fan at first, so it does not tip over. As the "bubble" starts to inflate, take a pair of scissors and cut a slit about 3 feet long in another side of the "bubble". Avoid cutting directly across from the box fan. This will prevent the bubble from lifting off the floor. It also serves as a door to get inside if students wish to explore.

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OMG! Missiles are Scattered Everywhere
The graph below shows the weight and the diameters of the Fleet Ballistic Missiles for the US Navy Submarines. Use the scatter plot to answer the questions found below.
 How many missiles have a diameter of 54 inches?
3 missiles
How did you determine the number? Count the number of dots above 54 on the horizontal (diameter) axis
 What does the point labeled "A" represent?
Point A represents a weight of 73,000 pounds and a diameter of 74 inches
How did you decide what it represented? These values are determined by reading the ycoordinate and the xcoordinate
 Using the pattern that exists with this data, if the US Navy developed a new missile that was approximately 100,000 pounds, what would its diameter be?
About 80 inches
Explain your answer. Imagine a line going through the data and then find the horizontal coordinate of the point associated with 100,000 pounds
 Suppose a missile had a diameter of 65 inches, what do you think the weight would be?
About 52,000 pounds
Explain your answer. Imaging a line going through the data and then find the y coordinate of the point associated with 65 inches in diameter
 Suppose the US Navy made a new and improved missile that weighed approximately 150,000 pounds, what do you think the diameter would be?
About 98 inches in diameter
Explain your answer. Answers may vary. Except all if reasonable, not just guesses.
 Describe the relationship between a missile's weight in pounds and its diameter in inches.
As the diameter increases, the weight increases so the relationship is nearly linear.

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Massive Missile Insides
The figure below shows the dimensions of the 6 Fleet Ballistic Missiles.
 Diameter for the Polaris A1, A2 and A3 is 54 inches
 Diameter for the Poseidon and Trident I is 74 inches
 Diameter for the Trident II is 84 inches
The volumes of the ballistic missiles are calculated using the lengths of the motors and the radius. The motor of the 3rd stages of the 3 latest missiles is about onethird of the diameter, so in order to find the volume of the 3rd stage divide the diameter by 3, then calculate radius and the volume of the 3rd stage. The volume of each motor is then added to get a total volume for each missile.
Use the formula for the volume of a cylinder to find the volumes of each missile motor section. The formula for the volume of a cylinder is V=Πr^{2}h, where V is the volume, r is the radius of the cylinder and h is the height of the cylinder.
Using the information above calculate the volume of each missile.
Calculations for Missile Volumes
Using the formula for the volume of a cylinder V=Πr^{2}h, these calculations were made:
Polaris A1^{ } 
volume of stages 1 and 2 combined. V=Π(2.25)^{2}(14) = 70.875Π 
Polaris A2^{ } 
volume of stages 1 and 2 combined. V=Π(2.25)^{2} (17) = 86.0625Π 
Polaris A3^{ } 
volume of stages 1 and 2 combined. V=Π(2.25)^{2}(18) = 91.125Π 
Poseidon C3^{ } 
volume of stages 1 and 2 combined. V=Π(3.08)^{2}(20) = 189.728Π
volume of stage 3. V=Π(2.06)^{2}(8) = 33.9488Π
volume of stages 1, 2 and 3 combined. V=189.728Π+33.9488Π = 223.6768Π 
Trident I C4^{ } 
volume of stages 1 and 2 combined. V=Π(3.08)^{2}(20) = 189.728Π
volume of stage 3. V=Π(2.06)^{2}(8) = 33.9488Π
volume of stages 1, 2 and 3 combined. V=189.728Π+33.9488Π = 223.6768Π 
Trident II D5^{ } 
volume of stages 1 and 2 combined. V=Π(3.5)^{2}(29) = 355.25Π
volume of stage 3. V=Π(2.33)^{2}(8) = 43.4312Π
volume of stages 1, 2 and 3 combined. V=355.25Π+43.4312Π = 398.6812Π 
** Remember that in order to find the 3rd stage radius, first divide the diameters by 3, then by 2. 

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How Far will a Missile Go?
What does the volume of a submarine missile have to do with the distance it travels?
Below is a table showing the total missile volumes and their ranges for the six U.S. Navy Fleet Ballistic Missiles.
Use the table above to complete the questions below:
 Using Grid paper make a scatter plot to display the relationship between total missile volume and the range of the missile. Describe the relationship you see.
 Draw a "best fit" line through the points on the graph. Use the line to find the equation for the line of fit. Write the equation for the line in slopeintercept form.
y = 4.921x + 592.652 (This is the calculator generated equation. The equations that are configured by hand may be close but probably not exact.)
 Use the graph of the line and the equation of the line to identify the slope and the yintercept. Explain what these values tell about the volume and the range of the missiles.
The slope is 4.921 and the yintercept is 592.652. The slope tells me how much the volume changes with the addition of each mile. The yintercept tells me what the range would be if the volume of the missile is 0, which does not make sense in this real world situation.
 Use your equation to find the range of the missile that has a volume of 500 cubic feet.
~3053 miles
 Use your equation to find the range of the missile that has a volume of 1500 cubic feet.
~7973.8 miles
 Use your equation to find the volume of a missile that traveles a range of 3500 miles.
~591 cubic feet
 Use your equation to find the volume of a missile that traveles a range of 800 miles.
~43 cubic feet
 Using Technology: Using the graphing calculator, enter the data points into the list menu.
 Use the calculator to determine the equation that will best describe your data. Compare it to the equation you developed in question #2.
 Using the table set and table key options on the calculator, find the range for a volume 500 cubic feet and 1500 cubic feet.
 Using the table key option on the calculator, find the volume of a missile that traveles 3500 miles; and 800 miles.



