- #1
emjay66
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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 [itex]V_b[/itex] = vector representing the direction of the boat (pointing north), [itex]V_a[/itex] = 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 [itex]V_f[/itex]). I assume the angle between [itex]V_b[/itex] and [itex]V_f[/itex] is 45 degrees (as this represents the direction of the pennant when the sailboat is underway). So using the sine rule I get
[itex] \frac{10}{\sin 45} = \frac{V_f}{\sin 120} [/itex]
which means [itex]x = 12.25[/itex]
Using the cosine rule to calculate [itex]V_a[/itex] results in [itex]V_a = 3.66km/hr[/itex] which does not match the answer in the book , which [itex]2.7km/hr[/itex]. 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 [itex]8.96 km/hr[/itex]
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.
(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 [itex]V_b[/itex] = vector representing the direction of the boat (pointing north), [itex]V_a[/itex] = 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 [itex]V_f[/itex]). I assume the angle between [itex]V_b[/itex] and [itex]V_f[/itex] is 45 degrees (as this represents the direction of the pennant when the sailboat is underway). So using the sine rule I get
[itex] \frac{10}{\sin 45} = \frac{V_f}{\sin 120} [/itex]
which means [itex]x = 12.25[/itex]
Using the cosine rule to calculate [itex]V_a[/itex] results in [itex]V_a = 3.66km/hr[/itex] which does not match the answer in the book , which [itex]2.7km/hr[/itex]. 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 [itex]8.96 km/hr[/itex]
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.