Find the velocity of the wind relative to the boat

[gnits]

Homework Statement

Find velocity of wind relative to boat

Relevant Equations

Va-Vb=Vab

Could someone please help me see where I am going wrong here?

During a race between two boats A and B there is a wind of 18 km/h blowing from due north. The resultant velocity if A is 12 km/h on a bearing of 060. Find the direction of the wind relative to A.

My reasoning:

(Relative to the Earth) A is traveling on a vector $12\,sin(60) i +12\,cos(60) j = 6\sqrt{3}i + 6j$

(Relative to the Earth) The wind is traveling on a vector $-18j$

So from A's point of view, vertically, the wind will seem to be coming in a direction of $-18-6=-24i$ and horizontally from a direction of $0-6\sqrt{3}i=-6\sqrt{3}i$

And so from A's point of view the wind will be coming from a direction of $270 - tan^{-1}(-24\,/\,(6\sqrt{3}))=270-66.59=203.41$

Book's answer is 193.9.

Thanks for any help,
Mitch.

 

[PeroK]

Which directions are $i$ and $j$ relative to North?

 

[gnits]

i is North and j is East

 

[PeroK]

i is North and j is East

Unless I'm missing something, your answer looks correct. Although, with $i$ being East. You seem to have $i$ changing between East and North from one line to the next.

Why not use E and N for East and North?

 

[Cutter Ketch]

So from A's point of view, vertically, the wind will seem to be coming in a direction of −18−6=−24i−18−6=−24i-18-6=-24i and horizontally from a direction of 0−6√3i=−6√3i0−63i=−63i0-6\sqrt{3}i=-6\sqrt{3}i

I think there is much confusion and imprecision in this statement. First, does “coming in a direction” mean “coming from a direction” or “blowing in a direction”? Those are opposite things. You’ve mixed them. Which do you mean? You use -18. With the negative sign that suggests you mean the direction the wind is traveling toward, not the direction it is coming from. (Also note that you used “i” where you meant “j”.) To this you add -6. This suggests that you believe that the apparent wind is opposite the direction of travel of the boat. (correct). Ok, if you meant the direction the wind is traveling toward, and you meant to use a j instead of an i, so far so good. However in the horizontal part directly below you say “the wind is coming from” but then use the negative sign indicating that you think the apparent wind is in the same direction as the boat travels (incorrect). So, which is it? I know it may sound like I’m nit picking, but this stuff is important. You will confuse yourself.

 

[neilparker62]

203.41 degrees looks good using cosine rule for Va , then sine rule for bearing of wind relative to boat.

 

[gnits]

Thanks everybody for your help, I really appreciate it, it was very helpful.

Mitch.