根据图片内容创建视频---**Extraction Content:**
**Section: 问题简化 (Problem Simplification)**
* **Title:** 问题简化 (Problem Simplification)
* **Problem Description:** 帆船在海面上乘风远航,确定最佳的航行方向及帆的朝向。(Sailing on the sea with the wind for a long voyage, determine the best sailing direction and the orientation of the sail.)
* **Simplified Problem Title:** 简化问题 (Simplified Problem)
* **Simplified Problem Description:** 海面上东风劲吹,设帆船要从A点驶向正东方的B点,确定起航时的航向口 (Strong east wind on the sea, assume the sailboat needs to sail from point A to point B, which is directly east of A, determine the initial sailing direction.)
* **Diagram 1 Description:**
* Type: Direction/Vector diagram.
* Elements:
* Points: A and B are labeled on a horizontal line.
* Line A-B: Represents the direction from A to B, labeled implicitly as 正东方 (East).
* Direction Indicator: An arrow pointing upwards, labeled 北 (North).
* Wind Direction: An arrow pointing from right to left, labeled 风向 (Wind direction), indicating East wind.
* Sailboat Icon: A small image of a sailboat.
* Sailboat Orientation Line: A line segment representing the orientation of the sailboat.
* Sailing Direction: An arrow originating from A, pointing diagonally upwards and right, labeled 航向 (Sailing direction).
* Angle: An arc with the symbol 69 indicating the angle between the sailboat orientation line and the wind direction arrow.
**Section: 模型分析 (Model Analysis)**
* **Title:** 模型分析 (Model Analysis)
* **Text:** 风(通过帆)对船的推力 (Thrust of the wind (through the sail) on the boat)
* **Text:** 帆对船体部分的阻力 (Resistance of the sail on the hull part)
* **Text:** 推力w的分解 (Decomposition of thrust w)
* **Formula/Equation:** w=w1+w₂ (Vector sum)
* **Formula/Equation:** w₁=f₁+f₂ (Vector sum)
* **Text:** f1~航行方向的推力 (f1 ~ Thrust in the sailing direction)
* **Text:** 阻力p的分解 (Decomposition of resistance p)
* **Formula/Equation:** p=p₁+p₂ (Vector sum)
* **Text:** p1~航行方向的阻力 (p1 ~ Resistance in the sailing direction)
* **Diagram 2 Description:**
* Type: Force vector diagram.
* Elements:
* Origin Point: A common origin for force vectors.
* Vectors:
* w: Labeled "w", appears to be the resultant wind force.
* w1: Labeled "w1", a component of w, generally horizontal right.
* w2: Labeled "w2", a component of w, generally downwards/left. w is the vector sum of w1 and w2, forming a parallelogram.
* f1: Labeled "f1", along the same direction as w1, shown as a component of w1.
* f2: Labeled "f2", perpendicular to f1, shown as a component of w1. w1 is the vector sum of f1 and f2, forming a right-angled decomposition (indicated by a square symbol).
* p: Labeled "p" (pink color), appears to be the total resistance force, generally upwards/left.
* p1: Labeled "p1" (pink color), a component of p, generally horizontal left. p1 is shown opposite the direction of f1/w1.
* p2: Labeled "p2" (pink color), a component of p, generally upwards/right. p is the vector sum of p1 and p2, forming a right-angled decomposition (indicated by a square symbol).
* Angle: An arc with the symbol 69 indicating an angle, likely between the wind force vector w and the direction of f1/w1 (sailing direction).
* Right Angles: Square symbols indicating right angles between f1 and f2, and between p1 and p2.
**Section: 模型假设 (Model Assumptions)**
* **Title:** 模型假设 (Model Assumptions)
* **Assumptions:**
* w与帆迎风面积s1成正比. (w is proportional to the effective wind area s1 of the sail.)
* p与船迎风面积s2成正比, 比例系数相同且s1远大于s2. (p is proportional to the effective wind area s2 of the boat, the proportionality coefficient is the same and s1 is much greater than s2.)
* w2与帆面平行, 可忽略. (w2 is parallel to the sail surface, can be ignored.)
* f2, p2垂直于船身, 可由舵抵消. (f2, p2 are perpendicular to the hull, can be canceled out by the rudder.)
* 航向速度v与净推力f=f1-p1成正比. (Sailing speed v is proportional to the net thrust f=f1-p1.)
```
模型建立
$w=ks_1$
$p=ks_2$
$w_1=w\sin(\theta-a)$
$f_1=w_1\sin a = w\sin a \sin(\theta-a)$
$p_1=p\cos\theta$
$v=k_1(f_1-p_1)$
船在正东方向速度分量 $v_1=v\cos\theta$
• $w$ 与帆迎风面积 $s_1$ 成正比, $p$ 与船迎风面积 $s_2$ 成正比, 比例系数相同且 $s_1$ 远大于 $s_2$.
Chart/Diagram Description:
Type: Vector diagram illustrating forces and velocities on a sailing boat.
Elements:
- A stylized shape representing a boat, oriented generally upwards and slightly right.
- Vector $v$ pointing in the direction of the boat's movement.
- A pink line representing the orientation of the sail.
- Angle $a$ shown between the direction of the boat ($v$) and the sail line.
- Vector $w$ pointing horizontally to the right, representing wind direction.
- Angle $\theta$ shown between the horizontal direction and the sail line.
- Vector $w_1$ originating near the sail, pointing upwards and right, representing wind force component perpendicular to the wind. It makes an angle $\theta-a$ with the wind direction $w$.
- Vector $w_2$ shown as a component of $w_1$ parallel to the wind direction.
- Vector $f_1$ originating from the sail line, perpendicular to the sail line, pointing forward relative to the boat.
- Vector $f_2$ shown as a component of $w_1$ parallel to the sail line.
- Vector $p$ pointing horizontally to the left, representing resistance.
- Vector $p_1$ pointing backwards along the direction of $v$. It makes an angle $\theta$ with the horizontal direction (opposite to $w$).
- Vector $p_2$ shown as a component of $p$ perpendicular to the direction of $v$.
- Vector $v_1$ pointing horizontally to the right, representing the velocity component in the East direction.
- Angle $\theta$ is shown between the direction of the boat ($v$) and the horizontal direction ($v_1$).
模型建立
$v_1=v\cos\theta = k_1(f_1-p_1)\cos\theta$
$f_1=w_1\sin a = w\sin a \sin(\theta-a)$
$p_1=p\cos\theta$
模型求解
求 $\theta, a$, 使 $v_1$ 最大
1) 当 $\theta$ 固定时求 $a$ 使 $f_1$ 最大
$f_1=w[\cos(\theta-2a)-\cos\theta]/2$
$\Rightarrow a=\theta/2$ 时 $f_1=w(1-\cos\theta)/2$ 最大
2) 令 $a=\theta/2$,
$v_1=k_1[w(1-\cos\theta)/2 - p\cos\theta]\cos\theta$
求 $\theta$ 使 $v_1$ 最大 ($w=ks_1, p=ks_2$)
模型求解
$v_1=k_1[w(1-\cos\theta)/2 - p\cos\theta]\cos\theta$
$\theta=(k_1w/2)[1-(1+2p/w)\cos\theta]\cos\theta$ (Note: This line appears to be miswritten in the source image, possibly intended as $v_1 = ...$)
$w=ks_1, p=ks_2$
记 $t=1+2s_2/s_1$, $k_2=k_1w/2$
$v_1=k_2(1-t\cos\theta)\cos\theta = k_2[t - (\cos\theta - \frac{1}{2t})^2]$ (Note: This equation also appears potentially miswritten in the source image for the form shown on the right side)
$\Rightarrow \cos\theta = \frac{1}{2t} (t=1+\frac{2s_2}{s_1})$, $a = \frac{\theta}{2}$ $v_1$ 最大
$s_1 \gg s_2 \Rightarrow 1 < t < 2 \Rightarrow 1/4 < \cos\theta < 1/2 \Rightarrow 60^\circ < \theta < 75^\circ$
备 注
• 只讨论起航时的航向，是静态模型。
• 航行过程中终点B将不在正东方，应调整 $\theta$ 和 $a$。
```
Title: “扬帆远航”思考题
Questions:
(1)试想，如果把帆调整到与航向偏西北夹角α，其作用如何变化？
(2)如果一直刮东风，船的后半程（驶向东南方向）如何布置帆？画出草图并做力学分析。
(3)认真调研风帆产生的垂直船体的分力靠什么抵消的？
(4)本例中航向与风向的方向夹角大于90°，倾向于“倒推”，如果方向夹角小于90°，又该如何布置帆的方向？
(5)固定一下参数，本例中取θ=60°，忽视风给船体带来的直接阻力；帆的面积取100m²。重新用作图法(α∈[0°,60°])和极值点法求夹角α？
Annotation: 16