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"The flag of a ship that is moving northward with 10 km/h, points exactly southwestward. The windsock at the lighthouse points under 30 relative to the western direction southward. Calculate the velocity of the wind at land and on board of the ship".

I have done my homework and solved this problem, so this is not a homework question, but a conceptual one about flags. First, I have assumed that the ship velocity given in the problem is relative to the earth (that is the velocity as viewed by an observer on the earth). Second that the vector sum of the the wind velocity and the ship velocity relative to the wind equals to the ship velocity relative to the earth.

These assumptions all make sense to me, while I am a bit uncertain about the third assumption I needed to do in order to solve the problem: the indication given by the flag on the ship! My understanding is that when you have a flag on an object moving in the wind, this flag - as seen from the earth - waves in the direction opposite of that of the velocity of the object relative to the wind (that is the velocity produced by the ship's engines in this case). My question: Could someone please explain if this correct and why it is so?

In this problem this means that this latter velocity makes an angle of 45 degrees northeastward, the same angle made by the flag southwestward (which is the opposite direction in words).

My results are: speed of the wind: 27.32 km/h relative to earth, but 33.46 km/h relative to the boat. The latter higher value I got makes sense, intuitively speaking, since the wind is a headwind, so it is stronger relative to the ship!

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The trailing edge of the flag points directly in the direction of the velocity vector of the relative wind. In common terminology, an air mass whose velocity vector is South is called a North Wind, because if you face North you seem to be facing its source. But don't let that confuse you. – Mike Dunlavey Oct 12 '12 at 14:43
@Mike suppose the wind is 5 knots and your nuclear=powered ship is travelling at 50 knots downwind. Won't the flag point the opposite direction than if the ship were a sailing ship travelling at 2 knots? – RedGrittyBrick Oct 12 '12 at 14:55
@Red: Sure. It depends on the relative wind. 50 South - 5 South = 45 South. 2 South - 5 South = -3 South = 3 North. If boat speed downwind = wind speed downwind, the relative wind is 0. – Mike Dunlavey Oct 12 '12 at 19:25

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