5.16 两次流离失所的后果

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  导致的两次流离失所

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  You are outside playing frisbee on a warm afternoon with your friends. You are trying to throw the frisbee to your friend, but unfortunately, the wind is blowing and keeps pushing your frisbee away from your friend and off course. If you are throwing the frisbee at 20 miles per hour at your friend who is due North of you, and the wind is blowing toward the East at 5 miles per hour, what is the actual trajectory of the frisbee?
  ::你在外面和朋友们在温暖的下午玩飞盘。你正试图把飞盘扔给朋友,但不幸的是,风正在吹,而且不断将飞盘推离朋友和航向。 如果你把飞盘按每小时20英里的速度扔到你北边的朋友,而风则以每小时5英里的速度向东吹,飞盘的实际轨迹是什么?

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  Finding the Resultant of Two Displacements
  ::寻找两个流离失所问题的后果

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  We can use vectors to find direction, velocity, and force of moving objects. In this section we will look at a few applications where we will use resultants of vectors to find speed, direction, and other quantities. A displacement is a distance considered as a vector.
  ::我们可以使用矢量来找到移动对象的方向、 速度和力。 在本节中, 我们将查看一些应用, 我们将使用矢量的结果来找到速度、 方向和其他数量 。 迁移是一个视距为矢量的矢量 。

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  If one is 10 ft away from a point, then any point at a radius of 10 ft from that point satisfies the condition. If one is 28 degrees to the east of north, then only one point satisfies this.
  ::如果一个点距离一个点10英尺,那么该点半径10英尺的任何点就满足了条件。如果一个点位于北部以东28度,那么只有一个点就满足了这个条件。

  

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  Let's take a look at some problems involving the of two displacements. 
  ::让我们来研究一下 涉及到两个流离失所的一些问题。

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  1. A cruise ship is traveling south at 22 mph. A wind is also blowing the ship eastward at 7 mph. What speed is the ship traveling at and in what direction is it moving?
  ::1. 一艘游轮在22海里处向南航行,风也向东吹7英里处。 驶往何方和驶向何方的轮船速度如何?

  

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  In order to find the direction and the speed the boat is traveling, we must find the resultant of the two vectors representing 22 mph south and 7 mph east. Since these two vectors form a right angle, we can use the Pythagorean Theorem and trigonometric ratios to find the magnitude and direction of the resultant vector.
  ::为了找到船行驶的方向和速度,我们必须找到代表南纬22mph和东经7mph的两个矢量的结果。由于这两个矢量构成一个正确的角度,我们可以使用Pytagoren Theorem和三角比来找到所产生矢量的大小和方向。

  

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  First, we will find the speed.
  ::首先,我们将找到速度。

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  $$\begin{array}{rl}{22}^2+7^2& =x^2\\ 533& =x^2\\ 23.1& =x\end{array}$$

  
  ::222+72=x2533=x223.1=x

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  The ship is traveling at a speed of 23.1mph.
  ::这艘船的飞行速度是23.1米

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  To find the direction, we will use tangent, since we know the opposite and adjacent sides of our triangle.
  ::为了找到方向,我们将使用正切点,因为我们知道三角形的对面和相邻两边。

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  $$\begin{array}{rl}\tan \theta & =\frac{7}{22}\\ {\tan }^{-1}\frac{7}{22}& ={17.7}^{\circ }\end{array}$$

  
  ::722tan-1722=17.7

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  The ship’s direction is $\begin{array}{r}S{17.7}^{\circ }E\end{array}$ .
  ::该船的方向是S17.7-E。

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  2. A hot air balloon is rising at a rate of 13 ft/sec, while a wind is blowing at a rate of 22 ft/sec. Find the speed at which the balloon is traveling as well as its angle of elevation.
  ::2. 热气球以13英尺/秒的速度上升,风速以22英尺/秒的速度吹起。

  

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  First, we will find the speed at which our balloon is rising. Since we have a right triangle, we can use the Pythagorean Theorem to find calculate the magnitude of the resultant.
  ::首先,我们会找到气球上升的速度。既然我们有一个正确的三角形,我们可以使用毕达哥伦理论来计算结果的大小。

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  $$\begin{array}{rl}{x}^2& ={13}^2+{22}^2\\ x^2& =653\\ x& =25.6ft/\sec \end{array}$$

  
  ::x2=132+222x2=653x=25.6英尺/秒

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  The balloon is traveling at rate of 25.6 feet per second.
  ::气球以每秒25.6英尺的速度行驶

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  To find the angle of elevation of the balloon, we need to find the angle it makes with the horizontal. We will find the $\begin{array}{r}\mathrm{\angle }A\end{array}$ in the triangle and then we will subtract it from $\begin{array}{r}{90}^{\circ }\end{array}$ .
  ::要找到气球的高度角, 我们需要找到它与水平角的角。 我们会在三角形中找到 A, 然后从 90 中减去 。

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  $$\begin{array}{rl}\tan A& =\frac{22}{13}\\ A& ={\tan }^{-1}\frac{22}{13}\\ A& ={59.4}^{\circ }\end{array}$$

  
  ::A=2213A=tan-12213A=59.4

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  Angle with the horizontal $\begin{array}{r}=90-59.4={30.6}^{\circ }\end{array}$ .
  ::水平为 90 - 59.4 = 30.6\\ 的角 。

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  The balloon has an angle of elevation of $\begin{array}{r}{30.6}^{\circ }\end{array}$ .
  ::气球的高度角为30.6。

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  Continuing on, find:
  ::继续, 查找 :

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  How far from the lift off point is the balloon in 2 hours? Assume constant rise and constant wind speed. (this is total displacement )
  ::离升降点还有多远? 气球在2小时之内会持续上升,风速也会持续上升。 (这是全部移位)

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  After two hours, the balloon will be 184,320 feet from the lift off point (25.6 ft/sec multiplied by 7200 seconds in two hours).
  ::两小时后,气球将离起重点184 320英尺(25.6英尺/秒乘以7200秒,两小时后)。

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  How far must the support crew travel on the ground to get under the balloon? ( horizontal displacement )
  ::支援人员必须到多远的地面才能到达气球下面?

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  After two hours, the horizontal displacement will be 158,400 feet (22ft/sec multiplied by 7200 seconds in two hours).
  ::两小时后,横向迁移将达158 400英尺(22英尺/秒乘以7 200秒,两小时)。

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  If the balloon stops rising after 2 hours and floats for another 2 hours, how far from the initial point is it at the end of the 4 hours? How far away does the crew have to go to be under the balloon when it lands?
  ::如果气球在2小时后停止上升,再漂浮2小时,离最初点还有多远? 4小时结束时,机组人员在气球落地时还要离气球多远?

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  After two hours, the balloon will have risen 93,600 feet. After an additional two hours of floating (horizontally only) in the 22ft/sec wind, the balloon will have traveled 316,800 feet horizontally (22ft/second times 14,400 seconds in four hours).
  ::两小时后,气球将上升93,600英尺,在22ft/sec风中再漂浮两小时(仅浮浮)后,气球将水平行驶316,800英尺(4小时内,22英尺/秒乘以14,400秒)。

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  We must recalculate our resultant vector using Pythagorean Theorem.
  ::我们必须使用毕达哥里安理论 重新计算我们产生的矢量

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  $$\begin{array}{r}x=\sqrt{{93600}^2+{316800}^2}=330338ft.\end{array}$$

  
  ::x=936002+3168002=330338英尺

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  The balloon is 330,338 feet from its initial point. The crew will have to travel 316,800 feet or 90 miles (horizontal displacement) to be under the balloon when it lands.
  ::该气球距离最初点330,338英尺,乘务人员必须行驶316,800英尺或90英里(横向移位)才能降落在气球下。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find the actual trajectory of the frisbee. 
  ::早些时候,你被要求 找到飞盘的实际轨迹。

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  When you draw each of the vectors, one for your throw of the frisbee and one for the wind, they look like this:
  ::当你画出每一个矢量, 一个是用来扔飞盘的, 一个是用来吹风的, 它们看起来是这样的:

  

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  And combining them, as we saw in this section, results in this:
  ::正如我们在本节所看到的那样,将它们结合起来,就会产生以下结果:

  

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  The diagonal line in above diagram is the result of both of the vector displacements, and is therefore the actual path of the frisbee.
  ::以上图表中的对角线是矢量迁移的结果,因此是飞盘的实际路径。

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  Example 2
  ::例2

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  Does $\begin{array}{r}|\stackrel{\to }{a}+\stackrel{\to }{b}|=|\stackrel{\to }{a}|+|\stackrel{\to }{b}|\end{array}$ ? Explain your answer.
  ::解释一下你的回答

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  When two vectors are summed, the magnitude of the resulting vector is almost always different than the sum of the magnitudes of the two initial vectors. The only times that $\begin{array}{r}|\stackrel{\to }{a}+\stackrel{\to }{b}|=|\stackrel{\to }{a}|+|\stackrel{\to }{b}|\end{array}$ would be true is when 1) the magnitude of at least one of the two vectors to be added is zero, or 2) both of the vectors to be added have the same direction.
  ::当对两个矢量进行总和时,所生成的矢量的大小几乎总是与两个初始矢量的大小之和不同。唯一真实的时间是 (1) 所添加的两个矢量中至少有一个的大小是零,或者 (2) 所添加的两种矢量的大小都具有相同的方向。

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  Example 3
  ::例3

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   A plane is traveling north at a speed of 225 mph while an easterly wind is blowing the plane west at 18 mph. What is the direction and the speed of the plane?
  ::飞机向北飞行的速度为225平方英里,而东风正以18平方英里向西吹风。 飞机的方向和速度是多少?

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  Speed (magnitude): $\begin{array}{r}\sqrt{{18}^2+{225}^2}=225.7\end{array}$ and its direction is $\begin{array}{r}\tan \theta =\frac{18}{225}=N{4.6}^{\circ }W\end{array}$
  ::速度(磁度): 182+2252=225.7,方向为 tan18225=N4.6W

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  Example 4
  ::例4

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  Two workers are pulling on ropes attached to a tree stump. One worker is pulling the stump east with 330 Newtons of forces while the second working is pulling the stump north with 410 Newtons of force. Find the magnitude and direction of the resultant force on the tree stump.
  ::一名工人用330牛顿的力量向东拉立木桩,而第二手则用410牛顿的力量向北拉立木桩。

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  The magnitude is $\begin{array}{r}\sqrt{{330}^2+{410}^2}=526.3\end{array}$ Newtons and the direction is $\begin{array}{r}{\tan }^{-1}\left(\frac{410}{330}\right)=E{51.2}^{\circ }N\end{array}$ .
  ::数值为3302+4102=526.3牛顿,方向为Tan-1(410330)=E51.2N。

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  Review
  ::回顾

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  A ship is traveling north at 30 mph. A constant 5 mph wind is coming from the east (blowing west).
  ::一艘船舶在30米时向北行驶,从东部(西吹)吹来5米时的恒定风。

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  1. What is the ship's actual speed?
     ::飞船的实际速度是多少?
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  2. In what direction is the ship moving?
     ::船向哪个方向移动?
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  3. How far does the ship travel in 2 hours?
     ::两小时后船能飞多远?
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  4. If the ship travels for 2 hours and then just floats for another two hours, how far has the trip traveled?
     ::如果这艘船飞行了两个小时 然后再漂浮了两个小时 旅行又走了多远?
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  5. If the ship travels for 2 hours and then just floats for another two hours, how far from its starting point is the ship?
     ::如果船行驶2小时,再漂浮2小时,船离起点有多远?

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  A balloon is rising at a rate of 20 cm/s, while a wind is blowing at a constant rate of 15 cm/s.
  ::气球以20厘米/秒的速度上升,而风速以15厘米/秒的恒定速度吹。

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  6. What is the balloon's actual speed?
     ::气球的实际速度是多少?
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  7. What is the balloon's angle of elevation?
     ::气球的高度角度是什么?
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  8. At what rate would the wind have to be blowing for the balloon to be traveling at 50 cm/s?
     ::气球以50厘米/秒的速度行驶 风能吹到什么程度?
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  9. After 3 hours, how far must the support crew travel on the ground to get under the balloon?
     ::3小时后 支援人员必须到多远 才能到达气球下面?

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  A plane is traveling west at a speed of 500 mph while a northerly wind is blowing the plane south at 30 mph.
  ::一架飞机正以500平方英尺的速度向西行驶,而北风在30平方英尺向南吹。

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  10. What is the actual speed of the plane?
      ::飞机的实际速度是多少?
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  11. What is the actual direction of the plane?
      ::飞机的实际方向是什么?
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  12. How far will the plane travel in 4 hours?
      ::四小时后飞机会飞多远?
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  13. How long will it take for the plane to travel exactly 3000 miles?
      ::飞机需要多久才能飞到3000英里?

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  A plane is takes off at a rate of 230 mph with an angle of 30° to the ground.
  ::飞机起飞速度为230英里,角度为30度至地面。

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  14. What is the rate at which the plane is receding from the ground?
      ::飞机从地面下降的速度是多少?
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  15. What is its ground speed?
      ::它的地面速度是多少?

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。