Protecting the fleet Part E. (Posted on 2015-12-23)

A fleet of ships is on a straight course at a speed of one knot. It is guarded by a cruiser that travels at two knots.

E. The ships are in the formation of a perfect circle of radius 1 nautical mile. How long does the cruiser take to completely traverse the edge of this circle?

Using
A to denote the angle moved by the cruiser round
the circle (seen relative to the fleet), where A = 0 gives
the direction of the fleet (x axis), and using v for the speed
ofthe cruiser relative to the fleet
after time t:
The velocity components are: Cruiser relative to the fleet:(-v sinA, v cosA) Fleet relative to the water:(1, 0)
So: Cruiser relative to water:(1 – v sinA, v cosA)

and to keep the cruiser’s speed at 2 knots,

(1 – v sinA)^{2} + (v
cosA)^{2} = 2^{2}

which simplifies to:v^{2}
– 2 sinA – 3 = 0

giving:v
= sinA + sqrt(3 + sin^{2}A)

Since the radius is 1, v = dA/dt and therefore time, t,
is the integral of 1/v with respect to A.

So the time for one complete circle is:

Integral_{A from 0 to 2pi} ofdA/( sinA + sqrt(3 + sin^{2}A))

This is an elliptic integral, so no exact answer is possible,
but my computer says 3.91323.. hours.