SECTION III 

 TWO COURSES FOR MANEUVERING UNIT, SINGLE COURSE FOR GUIDE (THE FICTITIOUS SHIP) 



In section II we dealt with a Fictitious Guide which, while maintaining a single course and speed, accompanied the actual 

 Guide for an indefinite time while the latter was on either its first or its second course. One feature of this Fictitious Guide 

 is that it maintains constant bearing with the real Guide, even when the two are not together. This is true because the two 

 units either met or else departed from a common point. The Fictitious Guide was necessary in the solution of the problems 

 listed in section II because we either did not know when the problem started or else we did not know when it terminated. 



In the cases to be investigated in this section, we do know when the problem starts as well as when it terminates. We 

 know, also, the course and speed of the Guide and the initial and final relative positions of the Maneuvering Unit. The 

 Maneuvering Unit, however, is not going directly from the initial position to the final position, but it is going to some inter- 

 mediate position en route. We therefore have two Lines of Relative Movement to consider instead of the one usually involved. 

 In some cases these Relative Movement Lines may be specified chart lines. Although we know the total time of the problem, 

 we do not know when the Maneuvering Unit must reach the intermediate point until we have partially solved the problem. 



To facilitate the solution of problems of this type, another fictitious unit is introduced. This unit is called the Ficititious 

 Ship and will be used in both surface and aerial problems. The Fictitious Ship leaves the initial relative position simultane- 

 ously with the Maneuvering Unit, proceeding directly for the final relative position at such a speed that the time of arrival 

 of the Ficititious Ship coincides with the time of arrival of the Maneuvering Unit. The latter, meanwhile, has passed through 

 the required intermediate point. 



Since the Fictitious Ship and the Maneuvering Unit leave the initial point at the same instant, they maintain constant 

 bearing while the Maneuvering Unit is proceeding to the intermediate point. Also, since these two units come together again 

 at the final position, they maintain constant bearing while the Maneuvering Unit is proceeding from the intermediate point 

 to the final position. The bearing of the Fictitious Ship from the Maneuvering Unit (and the reverse), therefore does not 

 change during the problem. It is this feature which gives the Fictitious Ship its particular value in the solution of two-course 

 problems. 



Another feature, intimately connected with the Fictitious Ship, is called the Time Line and is illustrated in figure 31, 

 which shows the Vector Diagram, the Relative Plot, and the Navigational Plot. In this sketch, both the Fictitious Ship 

 and the Maneuvering Unit are originally at the point A. The Maneuvering Unit steers course 050° for 2.0 hours at a speed 

 of 12.0 knots until point P is reached, when course is changed to 320° and speed to 18.0 knots. This latter course and speed 

 are continued for 1.0 hour, at the end of which time the Maneuvering Unit has arrived at point B. The Fictitious Ship, which 

 will be designated as F leaves point A and heads directly for point B, using course 013° and speed 10.0 knots. It will be 

 noted that F leaves point A and arrives at point B simultaneously with the Maneuvering Unit M. 



As M proceeds from point A to point P in 2.0 hours, F runs from A to point F x ; therefore, at the end of 2.0 hours, the bearing 

 of F from M is P . . . . Pi, and this bearing has not changed since the two units left point A. When M turns at point P, and 

 heads for point B, F continues along the line F t . . . . B. The bearing between M and F does not change, therefore, and the 

 two units would be in collision at point B. Using the Vector Diagram, where the vectors for the first and second legs run by 

 Maree .... mi and e . . . . m 2 respectively, the slope P .... F x when transferred to mi is found to also pass through m 2 and 

 /, the extremity of the vector for F. This, of course, is as it should be, because the bearing between F and M does not change 

 during the entire problem. 



At the end of 2.0 hours, M has departed a distance Fi '.'... P from F. This distance divided by the time involved yields a 

 rate which, when laid off on m x . . . . m 2 , is found to coincide with mi .... /. Since M must again coincide with F at the 

 end of the problem, or in 1 hour, the relative rate of approach between M and F must be P . . . . F t divided by 1.0 hour. 

 Laying off this rate from m 2 , we see that it indicates the vector m 2 . . . . /. From this, it is apparent that the line connecting 

 the two vectors of M also includes the end of F's vector, and that the rates m x . . . . j and m 2 . . . . / are directly proportional 

 to the rates of departure and approach of the two units and are inversely proportional, to the times of departure and approach. 

 The two units are departing from one another while M is on course e . . . . mi and approaching while M is on course e . . . . m 2 , 

 hence the line m x . . . . m 2 is divided by / into segments which are inversely proportional to the times M spends on the first 

 and the second courses, mi .... / and m 2 . . . . / are relative vectors because the two units are on the courses indicated 

 for M and for F simultaneously, mi .... m 2 is not a relative vector, but is what is described as a Time Line, so called because 

 of the above-mentioned time division determined by/. 



For convenience, the characteristics of the Fictitious Ship may be summarized as follows: 



1 . It proceeds at constant speed and on a single course, from the initial position to the final position, leaving the former and 

 arriving at the latter coincidentally with its reference ship, or unit, which reference ship or unit meanwhile passes through some 

 intermediate point. 



2. It maintains constant bearing with its reference unit throughout the operation. 



3. The Fictitious Vector divides the line connecting the two vectors of the Reference or Maneuvering Unit into segments 

 inversely proportional to the times spent by the Maneuvering Unit on its two courses. 



4. The Fictitious Ship can be used only for the solution of problems in which both time of start and time of finish are known. 



The above characteristics of the Fictitious Ship will become familiar during the solution of the two-course problems pre- 

 sented in illustrating the cases for this section. A number of scouting problems are included, involving both surface craft and 

 aircraft. During these problems when minimum speed is required, the earlier the Maneuvering Unit leaves its initial position 

 the lower will be the speed necessary. Also, to use minimum speed, the same speed must be used on both legs. 



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