Sailing Blind: The Challenges of a Submarine's Navigator
John Clark, Physics Teacher and Military Historian, Deltona HS, Deltona
2012 Naval Historical Foundation STEM-H Teacher Fellowship
Instructional Goal
How does a submariner navigate when submerged? Let your students find out as they explore the
concept of vector addition as used in the Navy to combine dead reckoning calculations and
inertial navigation data to determine the ships position after many hours under the sea. Can your
students navigate through the Strait of Gibraltar to reach a safe port?
Students will complete activities applying the concept of vector addition to a real world
application - a submariner calculating the ship's position while submerged.
Common Core State Standards:
CCSS HSN – VM.B.4 Perform operations on vectors.
CCSS.Math.Content.HSN.VM.B.4.a
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand
that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
CCSS.Math.Content.HSN.VM.B.4.b
Given two vectors in magnitude and direction form, determine the magnitude and
direction of their sum.
Background:
In the early days of sailing, navigating by the stars or celestial navigation kept ships on course. A
tool called a sextant allowed a sailor to calculate the ship's position with a fair degree of
accuracy. A sextant allowed the navigator to determine the angle between a selected star and the
horizon. The sextant measured the angle from the horizon up to the star. Using simple
trigonometry a possible range of positions could be determined. By determining the angle to
other stars, an accurate position can be found.
When sailing under the water getting a star fix is difficult as submarines have no windows and
when it surfaces it is highly vulnerable to detection. Avoiding detection remains a submarine's
greatest protection. Still, a submarine navigator must have a good sense of where the ship is even
when underwater and underway.
With advances in technology it has become easier for a submarine to know its current location
with high precision. Often, only putting up the periscope a few feet above the water line is
enough to get a location "fix" in today's world. Ironically, while the math is automated and the
data transfer lighting fast, the concept of today's state of the art GPS tracking is still based on the
same "triangulation" principles used with the sextant.
So how do you navigate a submarine underwater? The answer is a combination of two systems
that can calculate a ship's position - dead reckoning and inertial navigation. Dead reckoning uses
a fixed starting point and then adds the speed and direction of ship to determine the current
position. However going in a straight line in the ocean is difficult in comparison to dry land.
Currents and tides in the water are also moving the ship around while it sails along and if these
factors are not accounted for it is very likely that the navigator is not where he thinks he is.
To account for the factors of current, tides, and not steering straight, the navigator uses additional
information collected from the inertial guidance system which collects data on forces acting on
the ship. If a current is pushing the ship, or there is a change in speed, or there are small course
changes, the inertial guidance system will collect that data for the navigator to use in adjusting
the position calculation predicted by dead reckoning. The inertial guidance system uses
gyroscopes and accelerometers to detect a ships movement. In a sense, the two devises sit in the
middle of the ship and the ship moves around it.
Resources:
Inertial Navigation systems - overview: www.usnavymuseum.org/Ex2_Navigation.asp.
Reading Navy Charts: Nautical Charts - "Reading Charts" 1999 US Navy Navigation Training.
Also at: https://www.youtube.com/watch?v=TEsQF-uk5qU (8:26)
Inertial Navigation systems - how they work: en.wikipedia.org/wiki/Inertial_navigation_system.
Gyroscopes explained: en.wikipedia.org/wiki/Gyroscopes.
Accelerometers explained: en.wikipedia.org/wiki/Accelerometer
Background on Celestial Navigation: en.wikipedia.org/wiki/Celestial_navigation.
Navy Satellite Navigation Video: This 1967 Navy film explains
the highly successful Navy Navigation Satellite System
(NAVSAT), which was built during the 1960's, and operated
through 1996. It includes a detailed explanation of the computing
equipment used to operate the system in the 1960's. Not quite
GPS, but great for navigating below and on the ocean sea lanes.
Finding the Strait of Gibraltar
Background:
Approximately 9 miles (14 km) wide at its narrowest point, the Strait of Gibraltar is the entry
point into the Mediterranean Sea from the Atlantic Ocean. It's bordered by the continents of
Africa and Europe, and the countries of Morocco to the south, and to the north Spain, the British
colony of Gibraltar, and the Spanish exclave of Ceuta.
It is the only way for a submarine assigned to patrol the Mediterranean Sea to transit from the
Atlantic Ocean. For a submerged submarine a 9 mile opening is considered very narrow and
dangerous to transit submerged due to the turbulent and depth dependant currents. However due
to the rough surface waters and extensive shipping traffic, the transit is commonly done
submerged.
Objective:
Use vector addition to adjust your dead reckoning location from where you think you are to
where you really are.
Instructions:
Students will work with two scenarios of data to calculate a final position. The first is for a basic
class and the second is for a more challenging assignment. The solution to the basic problem
follows the “advanced” problem.
Finding the Strait of Gibraltar (Basic) STUDENT WORK SHEET Name:______________
DATA:
You are the navigator of a submarine that has been submerged for 12 hours and traveling at flank
speed (20 knots) due east (090) toward the Strait of Gibraltar. Due to safety and security
concerns you must bring your ship through the strait without surfacing to refine your location.
“X” makes your current position based on dead reckoning and the most recent navigation “fix”.
Now, using vector addition you will calculate, using the information provided by the inertial
guidance system, and adjust your position on the chart.
1. Where are you? What will happen if you keep going straight east? What course do you need
to follow to get to the front of the Strait before steering directly to the east to “line-up” for the
straight transit, shown by the circle on the chart marked
.
2. Your scenario: dead reckoning – 20 knots for 12 hours puts you at “X”
3. Information from you inertial guidance system.
a) Fluid friction has held down your speed by 0.2 knots
b) Ocean currents pushed your ship 2 knots east and 3 knots south for seven hours
c) For three hours a bad storm pushed the ship south at 0.5 knots
Each grid square is two by two nautical miles. One knot equals on nautical mile per hour.
Use the grid chart on the next page for your solution, north is up (000/360), south is down (180),
east is right (090) and west is left (270) on the grid.
Finding the Strait of Gibraltar (Basic). On the grid above Each grid square is two by two
nautical miles. One knot equals on nautical mile per hour.
Finding the Strait of Gibraltar (advanced) STUDENT WORK SHEET Name:_______________ You are the navigator of a submarine that has been submerged for 12 hours and traveling at 16 knots per hour due east (090) toward the Strait of Gibraltar. Due to safety and security concerns you must bring your ship through the strait without surfacing to refine your location. “X” makes your current position based on dead reckoning and your most recent “fix”. Now, using vector addition calculate, using the information provided by the inertial guidance system, and adjust your position on the chart. Where are you? What will happen if you keep going straight east? What course do you need to follow in degrees and for how long at a speed of 4 knots to get to the front of the Strait before lining up on course 090 to the east (at the point shown by the circle on the chart.) Your scenario: dead reckoning – 16 knots for 12 hours puts you at “X”. Fluid friction slows your speed by 0.1 knots. Each grid square is two by two nautical miles. One knot equals one nautical mile per hour. Information from you inertial guidance system. a) currents pushed the ship 15 degrees north for 3 hours, 25 degrees north for 1 hour and 5 degrees south for 8 hours. b) The officer of the deck of the previous watch turned the ship 5 degrees south after the 8th hour. Information from you inertial guidance system. a) b) c) d) e) Fluid friction slows your speed by 0.1 knots. currents pushed the ship: 15 degrees north for 3 hours, 25 degrees north for 1 hour and 5 degrees south for 8 hours The officer of the deck of the previous watch turned the ship 5 degrees south after the 8th hour. Information from you inertial guidance system. 0.1 kt x 12 hr = 1.2 nm west 16kt x sin(15) x 3 hr = 12.42 nm c d 16kt x sin(25) x 1 hr = 6.76 nm -­‐16 kt x sin(5) x 8 hr = -­‐11.16 nm b e a 3.84 nm -­‐16kt x sin(5) x 3 hr = -­‐ 4.18 nm 55 nm 53.2 nm East/West distance = 52 nm + 1.2 nm = 53.2 nm North/South position = 12.42 + 6.76 – 11.16 – 4.18 = 3.84 nm New course = 090 + tan-­‐1 (3.84/53.2) = 090 + 4 = 094 Distance to Entrance Point = 3.84/sin(4) = 55 nm Time to change to course 090 = 55 nm/16 kt = 3 hr 26 min Each grid square is two by two nautical miles. One knot equals one nautical mile per hour. a) Basic problem: fluid friction has held down your speed by 0.2 knots. 0.2kt x 12hr = 2.4 nm west b) Ocean currents pushed your ship 2 knots east and 3 knots south for seven hours 2kt x 7hr = 14 nm east, 3 kt x 7hr = 21 nm south c) For three hours a bad storm pushed the ship south at 0.5 knots. 0.5 kts x 3 hr = 1.5 nm -­‐2.4 nm + 14 nm = 11.6 nm east X = 52 nm – 11.6 nm = 40.4 nm to go a 21 nm = 1. 5 nm = 22.5 nm too far south b 40.4 nm b 22.5 nm 46.5 nm Tan Ө = 22.5/40.4 so Ө = 29 Course is 090 – 29 = 061 until the turn c Distance = 22.5 nm/sin 29 = 46.5 nm Time to turn east = 46.5 nm/20kts = 2 hours, 19 minutes