Case XVH-A 



LOCUS OF MEETING POINTS OF TWO SHIPS 



GIVEN: INITIAL RELATIVE POSITION AND SPEED OF EACH UNIT. 





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TO DETERMINE: AREA WITHIN WHICH THE TWO UNITS CAN MEET. 



Example. — A ship S, with 10.0 knots speed, is now located 60.0 miles bearing 180° from ship F, which has 20.0 knots speed. 



Required. — (a) Locus of meeting points for F and S, using given speeds, (b) Limiting courses open to F at its given 

 speed, (c) Corresponding courses for S at its given speed. (See fig. 21.) 



Procedure. — Plot the slower ship at any convenient place S and locate the faster ship at F. Extend the line F . . . . S 

 indefinitely. 



The two ships will meet earliest when they are headed directly for each other, for then the Relative Distance is being 

 traveled at their combined speeds. The latest time of meeting will similarly occur when both ships are on the same course 

 and the faster ship is headed directly for the slower ship, for then the Relative Speed is the difference between the two speeds. 

 The points of earliest meeting and of latest meeting are conveniently located by travel of the slower vessel since the distances 

 traveled are less. 



The point of earliest meeting, P u and the point of latest meeting, P 2 , are the extremities of the diameter of a circle whose 

 circumference marks the locus of positions at which the two vessels can meet when using the given speeds. In case the slower 

 vessel makes less than its given speed, the area enclosed by this circle represents the locus of meeting places. 



Original separation X Speed of S, 



By means of the formula, S . . . . Pi— ~ — = — r-r locate Pj. In this particular instance, both 



Speed of F plus Speed of S 



the time of earliest meeting and the distance S .... Pi may be found by using the Logarithmic Scale. 



_ _ . ', ,1 Original separation X Speed of S 



S . . . . P 2 is then equal to — ~ 5 — ^s — = 5 3 — f^— • 



Speed of F minus Speed of S 



The center of the circle, O, may be found by bisecting the line Pi .... P 2 or by the formula, 



_ _ S . ... Pi plus S . . . . P 2 . _ D 



S . . . . 0= -S-z minus S .... Pi. 



Tangents drawn from F to the locus circle drawn from O with radius O .... Pi are the limiting courses that F may take 

 when both ships use their given speeds. Corresponding courses for S are from S to the points of tangency, or S .... K' and 

 S .... K respectively. 



Answer. — (a) Circumference of a circle of radius 40.0 miles and center 20.0 miles bearing 180° from S. (b) Courses be- 

 tween 150° and 210°, measured clockwise, (c) 090° if F's course is 150°; 270° if F's course is 210°. 



NOTE. — In case the speed of F is equal to the speed of S, the radius of the locus circle becomes infinite, and a perpendicular erected at the 

 midpoint of S .... F becomes the required locus. 



In the event that F decides to steer a course between the tangents, such as F . . . . X', then S has a choice of 2 courses, S .... X and 

 S . . . . X' . In the event that S decides to steer some intermediate course such as S .... Y while F is moving along the line F .... X', the 

 time of meeting will be found by dividing the distance F .... Y by F's speed. S will then use a speed lower than 10.0 knots, found by dividing 

 the distance S .... Y by the time previously found. 



Once the required locus has been found, one ship decides on the course she is to steer and so notifies the other vessel, which must set her course 

 to reach that meeting point. 



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