Case IX 



TO FIND THE COURSE AND RELATIVE POSITION OF A VESSEL FROM THREE OR MORE BEARINGS, KNOWING 



THE VESSEL'S SPEED 



GIVEN: OWN COURSE AND SPEED, SPEED OF SECOND VESSEL, AND THREE OR MORE BEARINGS 

 WITH TIME INTERVAL BETWEEN BEARINGS. 



TO DETERMINE: COURSE OF SECOND VESSEL AND RELATIVE POSITION AT ANY SUBSEQUENT 

 TIME. 



Example. — A ship A on course 350°, speed 14.0 knots, is taking radio bearings of ship B, which latter vessel is known to be 

 making 10.0 knots. Bearings taken at 0813, 0858, and 0928, after being converted to true bearings, yield 01 7%°, 035°, and 

 050°, respectively. 



Required. — (a) Course of B. (b) Range and bearing of B from A at 1000. (See fig. 12.) 



Procedure. — Plot position of ranging ship A at any convenient point, and from this point lay out observed bearings 

 A . . . . pi, A . . . . p 2 , and A . . . . p 3 . 



At any point lay out a slope across the bearing lines so inclined that intercepts between bearings are proportional to the 

 time intervals between bearings. Methods of accomplishing this are shown on the following page. Letter this slope 

 P, .... P 2 .... P 3 . 



Lay out ranging ship's vector e .... a. Transfer slope Pi .... P 2 .... P 3 to a, cutting 10.0-knot speed circle at 

 bi and b 2 . B therefore has two possible courses represented by vectors e . . . . bi and e . . . . b 2 . 



Assuming B's vector to be e .... b lt by using the time between the first and third bearings (75 minutes) and the Relative 

 Speed a .... 6i on the Logarithmic Scale, the Relative Distance B would run in this period of time is easily found. This 

 Relative Distance By. ... B 3 is plotted between the 0813 and the 0928 bearings parallel to the slope P x . . . . P 3 . B z 

 represents the position of B relative to A at 0928 under the assumption that e . . . . bi represents B's vector. Similarly, if 

 e . . . . b 2 is assumed to be proper vector for B, its relative position at 0928 plots at 2?' 3 . 



To find the Relative Positions possible for B at 1000, the respective Relative Speeds for the total time of 107 minutes on 

 the Logarithmic Scale, give the Relative Distances run on courses shown by vectors e . . . . b x and e . . . . b 2 locating 

 B 4 and B\. 



Answer.— (a) 108° or 001^°. (b) 39.4 or 8.8 miles, bearing 067°. 



NOTE. — When the speed of the second vessel is less than the speed of the ranging vessel, two courses will be obtained for the second vessel 

 except when the transferred slope is tangent to the known speed circle. 



In case the bearing does not change, it will not be possible to determine the Relative Position of B by this method. 



All solutions obtained by bearings alone should be used with caution as a wide final error may result from a comparative small error in taking 

 one or more bearings. This is borne out by the Radian Rule that "An error of 1° is an error of 1 mile at a distance of 60 miles." 



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