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The Apparent Wind

https://isaacphysics.org/questions/maths_ch7_7_q2?board=183d113b-f0f4-46d3-82ff-77a567730851&stage=a_level

I got the answer to this part c of this question, but I would just like to double check my reasoning.

For part b:
F_u=F*(cos(gamma) * sin(beta) - sin(gamma) * cos(beta))

and when alpha_min occurs F_u is 0 (given in the question but am a little confused about why)
so when F_u = 0 using part b answer: beta = gamma.

and looking at figure 1, I just imagined that if the direction of motion of the sail, u, turned anticlockwise towards the speed of wind, w, i.e. alpha gets smaller, then beta would also get smaller, so i just presumed (guessed) that alpha=beta and got the right answer, but (clearly) I dont have the rigid reasoning of why this happens?

Could someone please explain the above comments - any help greatly appreciated.
Reply 1
Original post by mosaurlodon
https://isaacphysics.org/questions/maths_ch7_7_q2?board=183d113b-f0f4-46d3-82ff-77a567730851&stage=a_level
I got the answer to this part c of this question, but I would just like to double check my reasoning.
For part b:
F_u=F*(cos(gamma) * sin(beta) - sin(gamma) * cos(beta))
and when alpha_min occurs F_u is 0 (given in the question but am a little confused about why)
so when F_u = 0 using part b answer: beta = gamma.
and looking at figure 1, I just imagined that if the direction of motion of the sail, u, turned anticlockwise towards the speed of wind, w, i.e. alpha gets smaller, then beta would also get smaller, so i just presumed (guessed) that alpha=beta and got the right answer, but (clearly) I dont have the rigid reasoning of why this happens?
Could someone please explain the above comments - any help greatly appreciated.

Edit - mulling it over, Id guess the game is to show that for all boat speeds (with a heading alpha), then f_u is negative, then the boat will stall. So by varying the "length" (magnitude of) u, then beta can vary between alpha (when the sail is parallel to w with u=0) and 0 when u (and v) is large. The only way this can occur is if (see next paragraph) alpha is less than gamma.

Not that different from wht youve done, but I suspect they want you to "do" (spot) part b) with alpha instead of beta, then its just a simple reason about what inequality is to get an F_u which is positive and hence what the minimum value is.

No great knowledge of this, but having gamma constant etc, seems unrealistic.
(edited 2 months ago)

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