6.3 两维动力的保持

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 二维动力保护组织

  Collapse Expand

  二维动力保护组织

  

  播放段落

  In a game of billiards, it is important to be able to visualize collisions in two dimensions – the best players not only know where the target ball is going but also where the cue ball will end up.
  ::在台球游戏中,必须能够想象到两个维度的碰撞 — — 最佳球员不仅知道目标球的去向,而且知道提示球的结局。

  播放段落

  Conservation of Momentum in Two Dimensions
  ::二维动力保护组织

  播放段落

  Conservation of in all closed systems is valid, regardless of the directions of the objects before and after they collide. Most objects are not confined to a single line, like trains on a rail. Rather, many objects, like billiard balls or cars, can move in two dimensions. Conservation of momentum for these objects can also be calculated; momentum is a vector and collisions of objects in two dimensions can be represented by axial vector components. To review axial components, revisit : Resolving Vectors into Axial Components and Vectors: .
  ::在所有封闭系统中,无论物体相撞前后的方向如何,保护都是有效的。大多数物体不局限于一条线,如铁路上的火车。相反,许多物体,如台球或汽车,可以在两个维度上移动。也可以计算这些物体的动力;动力是一个矢量,两个维度的物体碰撞可以由轴矢量组件代表。要审查轴构件,请重新审视:将矢量溶入轴构件和矢量:

  播放段落

  Example 1  
  ::例1

  播放段落

  A 2.0 kg ball, $\begin{array}{r}A\end{array}$ , is moving with a of 5.00 m/s due west. It collides with a stationary ball, $\begin{array}{r}B\end{array}$ , also with a mass of 2.0 kg.  After the collision, ball  $\begin{array}{r}A\end{array}$ moves off at 30° south of west while ball  $\begin{array}{r}B\end{array}$ moves off at 60° north of west.  Find the velocities of both balls after the collision.
  ::2.0公斤球A正向西移动,向西移动5.00米/秒。它与固定球B相撞,质量为2.0公斤。碰撞后,A球在西南30度移动,B球在西北60度移动。在碰撞后找到两个球的速度。

  播放段落

  Since ball  $\begin{array}{r}B\end{array}$ is stationary before the collision, then the total momentum before the collision is equal to momentum of ball $\begin{array}{r}A\end{array}$ .  The momentum of ball  $\begin{array}{r}A\end{array}$ before collision is shown in red below, and can be calculated to be $\begin{array}{r}p=mv=(2.00\text{kg})(5.00\text{m/s})=10.0\text{kg m/s west}\end{array}$
  ::由于B球在碰撞前是静止的,因此碰撞前的总动量等于A球的动量。 A球在碰撞前的A球动量以下面红色显示,可以计算成Bep=mv=(2.00公斤)(5.00米/秒)=1.0万公斤/秒(西)

  

  播放段落

  Since momentum is conserved in this collision, the sum of the momenta of balls  $\begin{array}{r}A\end{array}$ and  $\begin{array}{r}B\end{array}$ after collsion must be 10.0 kg m/s west.
  ::由于这次碰撞保持了势头,相撞后A球和B球的瞬间总和必须是西经10.0千克/秒。

  播放段落

  $\begin{array}{r}{p}_{A\text{after}}=(10.0\text{kg m/s})(\cos {30}^{\circ })=(10.0\text{kg m/s})(0.866)=8.66\text{kg m/s}\end{array}$
  ::pafter = (10.0千克m/s)(cos30) = (10.0千克m/s)(0.866) = 8.66千克m/s

  播放段落

  $\begin{array}{r}{p}_{B\text{after}}=(10.0\text{kg m/s})(\cos {60}^{\circ })=(10.0\text{kg m/s})(0.500)=5.00\text{kg m/s}\end{array}$
  ::pafter = (10.0千克m/s)(cos60) = (10.0千克m/s)(0.500=5.00千克m/s)

  播放段落

  To find the final velocities of the two balls, we divide the momentum of each by its mass. Therefore,  $\begin{array}{r}{v}_A=4.3\text{m/s}\end{array}$ and $\begin{array}{r}{v}_B=2.5\text{m/s}\end{array}$ .
  ::为了找到两个球的最后速度, 我们将每个球的动力按质量分配。 因此, vA=4.3 m/s 和 vB=2.5 m/s 。

  播放段落

  Example 2  
  ::例2

  播放段落

  A 1325 kg car moving north at 27.0 m/s collides with a 2165 kg car moving east at 17.0 m/s.  The two cars stick together after the collision.  What is the and direction of the two cars after the collision?
  ::一辆1325公斤的汽车在27米/秒向北行驶,与一辆2 165公斤的汽车在17米/秒向东行驶。

  

  播放段落

  $\begin{array}{r}\text{Northward momentum}=(1325\text{kg})(27.0\text{m/s})=35800\text{kg m/s}\end{array}$
  ::北向动力=(1325公斤)(27,0米/秒)=35,800公斤/秒

  播放段落

  $\begin{array}{r}\text{Eastward momentum}=(2165\text{kg})(17.0\text{m/s})=36800\text{kg m/s}\end{array}$
  ::东向动力=(2165公斤)(170米/秒)=36800公斤/秒

  播放段落

  $\begin{array}{r}R=\sqrt{(35800{)}^2+(36800{)}^2}=51400\text{kg}\cdot \text{m/s}\end{array}$
  ::R=( 35800) 2+( 36800) 2=5140千克+s

  播放段落

  $\begin{array}{r}\theta ={\sin }^{-1}\frac{35800}{51400}={44}^{\circ }\text{north of east}\end{array}$
  ::东部以北=44 35800051400=44

  播放段落

  $\begin{array}{r}velocity=\frac{p}{m}=\frac{51400\text{kg}\cdot \text{m/s}}{3490\text{kg}}=14.7\text{m/s}@{44}^{\circ }\text{N of E}\end{array}$
  ::速度 = pm= 514000 kgm+m/s3490kg= 14.7m/s@ 44N of E

  播放段落

  Example 3  
  ::例3

  播放段落

  A 6.00 kg ball, $\begin{array}{r}A\end{array}$ , moving at velocity 3.00 m/s due east collides with a 6.00 kg ball, $\begin{array}{r}B\end{array}$ , at rest. After the collision,  $\begin{array}{r}A\end{array}$ moves off at 40.0° N of E and ball  $\begin{array}{r}B\end{array}$ moves off at 50.0° S of E.
  ::A6.00千克球,A,以速度3.00米/秒向东移动,休息时与6.00千克球相撞,B, 碰撞后,A在E°40.0度向下移动,B在E°50.0度向下移动。

  

  播放段落
  1. What is the momentum of  $\begin{array}{r}A\end{array}$ after the collision?
     ::碰撞后A的动力是什么?
  播放段落
  2. What is the momentum of  $\begin{array}{r}B\end{array}$ after the collision?
     ::相撞后B的动力是什么?
  播放段落
  3. What are the velocities of the two balls after the collision?
     ::碰撞后两个球的速率是多少?

  播放段落

    $\begin{array}{r}{p}_{\text{initial}}=mv=(6.00\text{kg})(3.00\text{m/s})=18.0\text{kg m/s}\end{array}$
  ::plip=mv=(6.00公斤)(3.00米/秒)=1.8万公斤/秒)

  播放段落

  This is a right triangle in which the initial momentum is the length of the hypotenuse and the two momenta after the collision are the legs of the triangle.
  ::这是一个右三角形, 最初的动力是下限的长度, 碰撞后的两个瞬间是三角形的腿。

  播放段落
  1. $\begin{array}{r}{p}_A=(18.0\text{kg m/s})(\cos {40.0}^{\circ })=(18.0\text{kg m/s})(0.766)=13.8\text{kg m/s}\end{array}$
     ::pA=(18.8千克m/s)(cos40.0)=(18.8千克m/s)(0.766)=13.8千克m/s
  播放段落
  2. $\begin{array}{r}{p}_B=(18.0\text{kg m/s})(\cos {50.0}^{\circ })=(18.0\text{kg m/s})(0.643)=11.6\text{kg m/s}\end{array}$
     ::pB=(18.8千克m/s)(cos50.0)=(18.8千克m/s)(0.643)=11.6千克m/s
  播放段落
  3. $\begin{array}{r}{v}_A=2.30\text{m/s}{v}_B=1.93\text{m/s}\end{array}$
     ::vA=2.30米/svB=1.93米/s

  播放段落

  Apply your understanding of the conservation of momentum in two dimensions to help Maria win a bocce ball game in this PLIX Interactive:
  ::运用你对在两个维度保持势头的理解, 帮助Maria在本次PLIX互动游戏中赢得比赛:

  播放段落

  Summary
  ::摘要

  播放段落
  • The conservation of momentum law holds for all closed systems regardless of the directions of the objects before and after they collide. 
    ::保护势头法适用于所有封闭系统,而不论碰撞前后物体的方向如何。
  播放段落
  • Momentum is a vector; collisions in two dimensions can be represented by axial vector components.
    ::动力是一种矢量;两个层面的碰撞可以由轴矢量构成。

  播放段落

  Review
  ::回顾

  播放段落
  1. Billiard ball $\begin{array}{r}A\end{array}$ , mass 0.17 kg, moving due east with a velocity of 4.0 m/s, strikes stationary billiard ball $\begin{array}{r}B\end{array}$ , also mass of 0.17 kg.  After the collision, ball  $\begin{array}{r}A\end{array}$ moves off at an angle of 30° north of east with a velocity of 3.5 m/s, and  ball  $\begin{array}{r}B\end{array}$  moves off at an angle of 60 ° south of east . What is the speed of  ball  $\begin{array}{r}B\end{array}$ ?
     ::Billiard Ball A,质量为0.17公斤,向东移动,速度为4.0米/秒,打击固定的Bliard Bball B,质量为0.17公斤,碰撞后,A球向东以北30度角移动,速度为3.5米/秒,B球向东以南60度角移动。B球的速度是多少?
  播放段落
  2. A bomb, originally sitting at rest, explodes and during the explosion breaks into four pieces of exactly 0.25 kg each. One piece flies due south at 10 m/s while another pieces flies due north at 10 m/s.
     播放段落
     1. What do we know about the directions of the other two pieces and how do we know it?
        ::我们对另外两块的方向了解多少? 我们怎么知道?
     播放段落
     2. What do we know about the speeds of the other two pieces and how do we know it?
        ::我们对另外两块的速度了解多少? 我们怎么知道?
     
     ::一个原坐着休息的炸弹爆炸,爆炸时破碎成四块,每块0.25公斤,其中一块在10米/秒向南飞,另一块在10米/秒向北飞,我们知道其他两块的方向和如何知道?我们知道其他两块的速度和如何知道?
  播放段落
  3. In a head-on collision between protons in a particle accelerator, three resultant particles were observed. All three of the resultant particles were moving to the right from the point of collision. The physicists conducting the experiment concluded there was at least one unseen particle moving to the left after the collision. Why did they conclude this?
     ::在粒子加速器质子之间的正面碰撞中,观察到了三个结果粒子。所有三个结果粒子都从碰撞点向右移动。进行实验的物理学家得出结论,在碰撞后至少有一个看不见的粒子向左移动。他们为什么得出这个结论?