One cold evening in the navy yard some southern officer, possibly desiring to heat up the wardroom before dinner, asked his shipmates when the mutual bearings between two ships in motion changed the fastest. If such was his purpose in bringing that up, he certainly accomplished his mission. Everybody had a definite notion and a different answer; none had ever worked it out.
We are brought up on the practical theorem that when a ship passes a fixed point on a constant course and at constant speed, the mutual bearings between the ship and the fixed point have their maximum rate of change when the ship has the point abeam. In this connection, too, it might be worth while to point out, incidentally, that the maximum “rate of change” takes place when the fixed point bears 45 degrees on the bow or on the quarter of the passing ship.
The simplest answer to both the problem of the two moving ships and the problem of the ship and the point is that the bearings change most rapidly when the intervening distance is the shortest. But if any practical use is to be made of the solution, there should be a more comprehensive answer, one that shows the relation between the respective speeds and the angular difference between the two courses. One might even ask, “When are the two ships closest together, anyway?”
In Fig. 1, suppose ships at A and B on courses as shown. Speeds are a and b respectively. As A moves to S, and B moves to T, LTSN, or LV, is change of bearing in unit time.
If LV is small and AB and ST are large, the line TN is the measure of LV.
Then V = TN = TM + MN — TM + PS = b sinF + a sinR
V will be maximum when its first differential equals zero.
Now since Z is constant, Y changes inversely as R, hence its differential will be negative with respect to the differential of R. Differentiating with respect to R,
dV/dR = —b cosF + a cosR = 0 (1) And V, the rate of change of bearing, will be largest when a cosR = bcosY (2)
Since R is the relative bearing of the second ship from the first, and Y is the relative bearing of the first ship form the second, and Z is the angular difference between their, R+Y+Z=180
In words, then, the rate of change of bearing between two moving points is maximum when the projections of their speed vectors upon the mutual line of bearing are equal.
In order to show that these are the conditions under which the distance between the two ships is minimum, it is necessary to show that the distance between the ships, both before and after they are in positions A and B respectively, is greater than AB.
Figure 2 is drawn to represent the travel of the ships for the unit time before reaching points A and B. One ship at L moves to A, and the other ship at C moves to B. In that time the change of bearing of the former from the latter is LX. At the end of that time a cosR = b cosY
KC is parallel and equal to AB.
But LC is greater than KC, hence LC is greater than AB.
Similarly, Fig. 1 represents conditions after the ships are in positions A and B. AB=SN and since ST is greater SN, ST is greater than AB.