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.
A. The fleet is travelling in a single file one nautical mile long. The cruiser starts at the back and continuously motors to the front and to the back again. How long does one cycle take?
B. The fleet is in a square formation, one nautical mile on a side, two sides parallel to the direction of motion. The cruiser hems close to the side of this square as it travels up one side, across the front, down the other side, and across the back. How long does one cycle take?
Try to give exact answers.
I will assume the fleet is traveling North.
When the cruiser is heading north its net speed relative to the fleet is 2-1 = 1 knot. Similarily when heading south, the cruiser's relative speed is 2+1=3 knots.
Since the line is one mile long, the cruiser's cycle time is 1/1 + 1/3 = 4/3 hours, or 1 hour and 20 minutes.
The two sides of the square formation that are parallel to the north-south axis take the same time as in part A.
On the other two sides the cruiser's speed is split into two vectors, one vector for the fleet's motion over the ocean and the other vector for the cruiser's movement across the fleet. These two vectors are perpendicular and their vector sum is the cruiser's 2 knot speed.
Let V be the cruiser's speed along the fleet. Then by Pythagoras 1^2 + V^2 = 2^2. Which simplifies to V=sqrt(3).
Then the cruiser's cycle time is 1/1 + 1/3 + 2*(1/sqrt(3)) = 4/3 + 2/sqrt(3) = 2.488034 hours, or approximately 2 hours, 29 minutes and 17 seconds.