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2 100 DEAD RECKONING 702. Plotting and Labeling the Course Line and Positions Draw a new course line whenever restarting the DR. Extend the course line from a fix in the direction of the ordered course. Above the course line place a capital C followed by the ordered course in degrees true. Below the course line, place a capital S followed by the speed in knots. Label all course lines and fixes immediately after plotting them because a conning officer or navigator can easily misinterpret an unlabeled line or position. Enclose a fix from two or more Lines of Position (LOP s) by a small circle and label it with the time to the nearest minute, written horizontally. Mark a DR position with a semicircle and the time, written diagonally. Mark an estimated position (EP) by a small square and the time, written horizontally. Determining an EP is covered later in this chapter. Express the time using four digits without punctuation, using either zone time or Greenwich Mean Time (GMT), according to procedure. Label the plot neatly, succinctly, and clearly. Figure 702. A course line with labels. Figure 702 illustrates this process. The navigator plots and labels the 0800 fix. The conning officer orders a course of 095 T and a speed of 15 knots. The navigator extends the course line from the 0800 fix in a direction of 095 T. He calculates that in one hour at 15 knots he will travel 15 nautical miles. He measures 15 nautical miles from the 0800 fix position along the course line and marks that point on the course line with a semicircle. He labels this DR with the time. Note that, by convention, he labels the fix time horizontally and the DR time diagonally. THE RULES OF DEAD RECKONING 703. Plotting the DR Plot the vessel s DR position: 1. At least every hour on the hour. 2. After every change of course or speed. 3. After every fix or running fix. 4. After plotting a single line of position. Figure 703 illustrates applying these rules. Clearing the harbor at 0900, the navigator obtains a last visual fix. This is called taking departure, and the position determined is called the departure. At the 0900 departure, the conning officer orders a course of 090 T and a speed of 10 knots. The navigator lays out the 090 T course line from the departure. At 1000, the navigator plots a DR position according to the rule requiring plotting a DR position at least every hour on the hour. At 1030, the conning officer orders a course change to 060 T. The navigator plots the 1030 DR position in accordance with the rule requiring plotting a DR position at every course and speed change. Note that the course line changes at 1030 to 060 T to conform to the new course. At 1100, the conning officer changes course back to 090 T. The navigator plots an 1100 DR due to the course change. Note that, regardless of the course change, an 1100 DR would have been required because of the every hour on the hour rule. Figure 703. A typical dead reckoning plot.

4 102 DEAD RECKONING Figure 705. Fix expansion. All possible positions of the ship lie between the lines tangent to the expanding circles. Examine this area for dangers. be 2 nm after the first hour, 4 nm after the second, and so on. At what value should the navigator start this error circle? Recall that a DR is laid out from every fix. All fix sources have a finite absolute accuracy, and the initial error circle should reflect that accuracy. Assume, for example, that a satellite navigation system has an accuracy of 0.5 nm. Then the initial error circle around that fix should be set at 0.5 nm. First, enclose the fix position in a circle, the radius of which is equal to the accuracy of the system used to obtain the fix. Next, lay out the ordered course and speed from the fix position. Then apply the fix expansion circle to the hourly DR s, increasing the radius of the circle by the error factor each time. In the example given above, the DR after one hour would be enclosed by a circle of radius 2.5 nm, after two hours 4.5 nm, and so on. Having encircled the four hour DR positions with the error circles, the navigator then draws two lines originating tangent to the original error circle and simultaneously tangent to the other error circles. The navigator then closely examines the area between the two tangent lines for hazards to navigation. This technique is illustrated in Figure 705. The fix expansion encompasses the total area in which the vessel could be located (as long as all sources of error are considered). If any hazards are indicated within the cone, the navigator should be especially alert for those dangers. If, for example, the fix expansion indicates that the vessel may be standing into shoal water, continuously monitor the fathometer. Similarly, if the fix expansion indicates that the vessel might be approaching a charted obstruction, post extra lookouts. The fix expansion may grow at such a rate that it becomes unwieldy. Obviously, if the fix expansion grows to cover too large an area, it has lost its usefulness as a tool for the navigator, and he should obtain a new fix by any available means. DETERMINING AN ESTIMATED POSITION An estimated position (EP) is a DR position corrected for the effects of leeway, steering error, and current. This section will briefly discuss the factors that cause the DR position to diverge from the vessel s actual position. It will then discuss calculating set and drift and applying these values to the DR to obtain an estimated position. It will also discuss determining the estimated course and speed made good Factors Affecting DR Position Accuracy Tidal current is the periodic horizontal movement of the water s surface caused by the tide-affecting gravitational forces of the Moon and Sun. Current is the horizontal movement of the sea surface caused by meteorological, oceanographic, or topographical effects. From whatever its source, the horizontal motion of the sea s surface is an important dynamic force acting on a vessel. Set refers to the current s direction, and drift refers to the current s speed. Leeway is the leeward motion of a vessel due to that component of the wind vector perpendicular to the vessel s track. Leeway and current combine to produce the most pronounced natural dynamic effects on a transiting vessel. Leeway especially affects sailing vessels and high-sided vessels. In addition to these natural forces, relatively small helmsman and steering compass error may combine to cause additional error in the DR.

6 104 DEAD RECKONING Figure 708b. Finding the course to steer at a given speed to make good a given course through a current. Figure 708c. Finding course to steer and speed to use to make good a given course and speed through the current. To find the course to steer and the speed to use to make good a desired course and speed, proceed as follows: See Figure 708c. Example 3: The captain desires to make good a course of 265 and a speed of 15 knots through a current having a set of 185 and a drift of 3 knots. Required: The course to steer and the speed to use. Solution: See Figure 708c. From A, any convenient point, draw AB in the direction of the course to be made good, 265 and for length equal to the speed to be made good, 15 knots. From A draw AC, the set and drift of the current. Draw a straight line from C to B. The direction of this line, 276, is the required course to steer; and the length, 14.8 knots, is the required speed. Answers: Course to steer 276, speed to use 14.8 kn.

Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation ED 5661 Mathematics & Navigation Teacher Institute August 2011 By Serena Gay Target: Precalculus (grades 11 or 12) Lesson

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