How Do You Solve Vector Problems in Physics?

[emjay66]

I am having trouble solving a basic problem in the use of vectors. The problem comes from Alonso & Finn "Fundamental University Physics" Volume 1, Chapter 3 Problem 12 and states "The pennant on the masthead of a sailboat streams back at an angle of 45 degrees (South of West), but the flag on the clubhouse extends out at 30 degrees south of west.
(a)If the speed of the boat is 10km/hr, find the wind velocity.
(b) find the apparent wind velocity for an observer on the boat."

My thinking so far has been the following:
For part (a) To calculate the wind speed I have assumed $V_b$ = vector representing the direction of the boat (pointing north), $V_a$ = wind speed (pointing 30 degrees north of east, as I am assuming that as the flag is pointing south of west and would be pushing the sailboat in a westerly direction, the sailboat would have to compensate by pointing in an easterly direction to sail north). The resulting vector would represent the direction the sailboat takes (call it $V_f$). I assume the angle between $V_b$ and $V_f$ is 45 degrees (as this represents the direction of the pennant when the sailboat is underway). So using the sine rule I get
$\frac{10}{\sin 45} = \frac{V_f}{\sin 120}$
which means $x = 12.25$
Using the cosine rule to calculate $V_a$ results in $V_a = 3.66km/hr$ which does not match the answer in the book , which $2.7km/hr$. Unfortunately this approach doesn't work for part (b) so I'm not really sure how to even approach part (b). (NB: the answer for part (b) is $8.96 km/hr$
Clearly I'm not understanding some aspect of this problem, so if someone can give me some broad hints as to where my thinking and understanding is incorrect that would be much appreciated.

 

[NascentOxygen]

Wow, it's easy to get confused here.

The wind is moving to the SW, so v_w is in that direction, 45 deg S of W.

The pennent shows v_w - v_b. (Check: If the wind is calm, the "apparent wind" is exactly opposite to the boat's velocity.)

So you know details of those two vectors.

How to find v_b?

 

[Delta2]

I am confused also with the terminology used here. 30 degrees south of west means that in the frame of reference of clubouse the wind vector forms 210 degree angle? Also the angle of the pennant (45 degrees south of west) is in the frame of reference of the boat or of the clubhouse?

 

[emjay66]

Delta2: Yes, As far as I see, the frame of reference (from due east) forms a 210 degree angle. Also, I assume that the angle of the pennant is in the frame of reference of the boat.

Sorry, the question doesn't provide more data (I copied the question (minus the diagrams) straight from the book)

 

[emjay66]

NascentOxygen: Your "Check" I actually made as a tacit assumption (ie no wind = "apparent wind" is exactly opposite the boats velocity)

However, I think I am still confused by your comment, that is, given the magnitude of the boats velocity ($v_b$) I'm not sure how $v_w -v_b$ would help me. Also, is the $v_w$ you are referring to the wind velocity with reference to the land or the boat, as they are different to each other?

 

[NascentOxygen]

However, I think I am still confused by your comment, that is, given the magnitude of the boats velocity ($v_b$) I'm not sure how $v_w -v_b$ would help me. Also, is the $v_w$ you are referring to the wind velocity with reference to the land or the boat, as they are different to each other?

My v's are vectors. The vectors are not relative to the boat, they are relative to the land (if a vector were relative to a moving body I would show that in a subscript because that detail is so important).

You know the direction of two vectors.

Hint: you will be looking to try to form a triangle because the closed triangle is our method for adding vectors.

 

[Delta2]

I am also try to solve this and i cant, i sense something is missing. If i understood correctly we work on the triangle of vectors $v_w,v_b,v_w-v_b$.

 

[NascentOxygen]

the answer for part (b) is $8.96 km/hr$

EDIT [strike]I don't get anything like either of the answers your book gives. But I can't see anything I'd change in my working.[/strike]

I've looked at this again, and now think there is not one answer, there are multiple. I'll try to work backwards from the text's answer to see if they have made some particular assumption.