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A differential equation is necessary to solve the problem of predicting
the location of two maneuvering ships because of the changes of course
they can make; only in an ideal situation will both of them be moving in
a straight line.
Consider
the following example. A firing ship is steaming course 000
(North) at 12 knots. Its target is steaming course 315 (Northwest)
at 12 knots. Initially, the relative bearing to the target ship is
090 (East) and the range 10,000 yards, just shy of 5 nautical miles
(4.937 nautical miles, to be more exact).
If
neither ship turns, then their relative positions five minutes later may
be calculated with trigonometry. The firing ship will have moved 1
nautical mile along course 000 and the target will have moved 1 nautical
mile along course 315. We know that it has therefore moved 0.707
(sin(45) = 0.707) nautical miles along course 270 (West) and 0.707
(cos(45) = 0.707) nautical miles along course 000. Therefore, the
distance between the firing ship’s track and the target ship has
decreased to 4.230 nautical miles and it has fallen behind by 0.293
nautical miles. These distances form the two sides of a right
triangle, with the hypotenuse being the linear distance between the two
ships.
Trigonometry
and the Pythagorean theorem again allow us to determine the length of
this hypotenuse and the angle of bearing of the target shi This distance
is 4.240 nautical miles ((0.293)2 + (4.230)2 =
(4.240)2), and the angle is 86 degrees (arctan(4.230/0.293) =
86). As this is the angle between the firing ship’s track and
the target, which is now slightly astern, the relative bearing becomes
094. The distance converts to 8,588 yards.
This
simplest of examples illustrates the basic principles involved in
plotting the movement of two ships. A significantly greater
magnitude of complexity would be introduced if the target ship were
turning to come onto a parallel course to the firing shi An
accurate calculation would then require a differential equation.
Effectively an infinite series of trigonometric calculations like the
example above would have to be calculated as the target ship traced
through the arc of its turn. It was exactly those sorts of
calculations that the Ford Rangekeeper was designed to solve.