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.
A. Relative to the fleet, the cruiser travels at 1 kt going forward and 3 kts going rearward. The round trip takes 1 hr + 1/3 hr = 1 hr, 20 min.
B. During the crosswise legs of the cruiser's motion, it is traveling along a diagonal, when seen from above, relative to the orthogonality defined by the fleet's square. In aerodynamical terms (hydrodynamical, I guess), there's a headwind (relative to the fleet) of 1 kt pushing the cruiser toward or away from the fleet's square. The 2-kt velocity of the cruiser must be directed so the shorter diagonal of the parallelogram of vectors is directed exactly sideways. The velocity relative to the fleet, v, then is given by:
v^2 + 1^2 = 2^2
v^2 = 4 - 1 = 3
as the resultant velocity is perpendicular to the velocity of the "headwind" relative to the fleet.
So the overall cycle takes 1 + 1/3 + 2/sqrt(3) ~= 2.48803387171258 hours.
Edited on December 15, 2015, 10:55 am
Posted by Charlie
on 2015-12-14 15:04:04