









Case XVII-B 





MEETING POINTS OF THREE SHIPS AT GIVEN SPEEDS 



GIVEN: POSITIONS AND SPEEDS OF THREE SHIPS THAT ARE TO MEET SIMULTANEOUSLY. 



TO DETERMINE: MEETING POINTS, COURSE FOR EACH SHIP, AND TIME OF MEETING. 



Example.— Ship A has ship B bearing 120°, distant 40.0 miles, and ship C bearing 030°, distant 25.0 miles. Speeds 

 available to the ships are as follows: A, 18.0 knots; B, 12.0 knots; and C, 16.0 knots. It is required that the three ships meet 

 simultaneously at their best speeds. 



Required. — (a) Location of earliest meeting point relative to A. (b) Course for each ship to earliest meeting place, 

 (c) Time for earliest meeting. (See fig. 22.) 



Procedure. — Plot the positions of the three ships at A, B, and C. 



By application of case XVII-A, determine the locus of meeting points of ships A and B, A and C, as well as ships B and 

 C. These loci intersect at K and K' . K, being closer to the initial positions of the three ships, is the earliest meeting point. 



Time for earliest meeting is found by dividing A .... K by speed of A, B .... K by speed of B, or C . . . . K by 

 speed of C. This is shown on the Logarithmic Scale. 



Answer.— (a) 31.5 miles bearing 088y 2 ° from A. (b) Course for A, O88V2 ; for B, 35iy 2 °; and for C, 137V 2 °. (c) 105 

 minutes. 







NOTE.— The following conditions must obtain if the three units are to be able to meet simultaneously: 



A times distance B . . . . C must be less than (B times distance A .... C plus C times distance A .... B) 

 B times distance A .... C must be less than (A times distance B . . . . C plus C times distance A .... B) 

 C times distance A .... B must be less than {A times distance B . . . . C plus B times distance A . . . . C), 

 where A is the speed of A; B, the speed of B; and C is the speed of C. If any of the above three relationships is an equality, there will 

 meeting point. 



Should the speeds available and initial relative positions be such that the above relationships do not exist, then the speed of the unit which 

 does not follow the above relationship must be changed. Assuming that A is the speed to be changed, the new speed lies between 



be but one 



(B times distance A . 



C) plus (C times distance A . . . . B) 



distance B 



where there is only one meeting place and 



(B times distance A 



C) minus (C times distance A . . . . B) 



distance B 



or A- 



(C times distance A 



. B) minus (B times distance A 



C) 



distance B 



where the relationships permit two meeting places. 



In the Navigational Plot, by v/hich this case is solved, it will be noticed that the centers of the locus circles all lie in a straight line. 



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