418 



Fishery Bulletin 90(2). 1992 



between vessel and dolphin at the point of closest 

 approach. 



Finally, to ease the task of data analysis by whoever 

 makes the necessary observations, we have derived the 

 expressions that relate dolphin speed and direction to 

 their range and bearing from the vessel. In a cartesian 

 coordinate system let V^ and Vy be, respectively, the 

 X and y components of the dolphin velocity minus, 

 respectively, the x and y components of the vessel 

 velocity. Both Vx and Vy are constructed using range 

 and bearing measurements of the dolphin from the 

 vessel. Then VD = [(Vx + VB)2 + Vy2]i'2 and a = arctan 



[Vy/(V, + VB)]. 



Problem solution 



As stated in the formulation above, the problem makes 

 sense only for the case Vd<Vb. With reference to 

 Figure 2, to maximize the distance between vessel and 

 dolphin at the point of closest approach, we must find 

 the maximimi value of r^^j^ with respect to angle < 



r» X 



'. actual track 

 '. of dolphin 



Figure 2 



The apparent motion of the dolphin as seen in the moving 

 frame of reference of the vessel. The apparent velocity V of 

 the dolphin is the resultant of the vector addition of -Vg 

 and Vp . Distance f is the perpendicular distance between the 

 initial position of the dolphin and the projected path of the 

 vessel, while a + f is the distance between vessel and dolphin 

 when the dolphin is abeam. 



For the purpose of the following exposition, we 

 define initial position to be that position of the vessel 

 (or the dolphin) at the time when the dolphin detects 

 the approaching vessel and begins evasion. Initial time 

 is the time corresponding to the initial position. 



In Figure 2, at the point of closest approach the 

 distance between vessel and dolphin is given by 



Tmin = (a + f)COS/J, 



(1) 



with respect to a simultaneously). 



In Figure 1, c is the distance along the projected path 

 of the vessel between the vessel's initial position and 

 the point that is abeam of the dolphin's initial position. 

 Let t(. be the time it takes the vessel to transverse 

 distance c, and t that time from the initial time until 

 the vessel has the dolphin abeam. From the applica- 

 tion of the Pythagorean Theorem we can deduce 



where 0</3<n/2, f is the perpendicular distance be- 

 tween the initial position of the dolphin and the pro- 

 jected path of the vessel, and (a-t-f) is the distance 

 between vessel and dolphin when the dolphin is abeam. 

 The vector diagram of Figure 2 shows that the ap- 

 parent track of the dolphin as seen from the vessel is 

 a function of a, the direction of escape of the dolphin. 

 Therefore, to solve the problem as posed, we must find 

 the extreme value of Eq. (1) with respect to angle a. 

 By computing the derivative with respect to a of Eq. 

 (1), we find that r^i„ is rendered an extreme value 

 when 



dfmin „^^ „ da 

 = cos p — 



da da 



(a-hf)sin/?^ 

 d« 



(2) 



vanishes. Depending on the functional dependence of 

 a and ft on a, an equation of the form of Eq. (2), could 

 vanish either term by term or by cancellation of the 

 terms. For the former case, each term could vanish 

 trivially (i.e., a and p are independent of a), or non- 

 trivially (i.e., both a and p are rendered extreme values 



Because 



(VDt)2 = a2 + [VB(t-t,)]^ 



a = Vpt sin a, 



(3) 



(4) 



it can then be shown by substitution of Eq. (4) into Eq. 

 (3) that 



*" (5) 



t = 



l-(VD/VB)cosa 



By substituting Eq. (5) into Eq. (4), we find that as a 

 function of a, a is given by 



a = 



Vote sin a 

 1-(Vd/Vb)cosc»" 



(6) 



Because at least a is a function of a, we can conclude 

 that in general Eq. (2) does not vanish trivially. How- 

 ever, p is also a function of a as can be deduced by the 

 application of the Law of Sines to Figure 2: 



