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?

Try to give an exact answer.

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
of  the 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 = 2²

which simplifies to:         v² – 2 sinA – 3 = 0

giving:              v = sinA + sqrt(3 + sin²A)

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} of  dA/( sinA + sqrt(3 + sin²A))

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

(compared with pi hours for a stationary fleet)

Posted by Harry on 2015-12-24 10:15:15