Flash/Flex学习笔记(43)：动量守恒与能量守恒

动能公式：​​​​

 

动量公式:​​​​

 

动量守恒：​​​​

 

能量守恒: ​​​​

 

根据这些规律可以得到下列方程组:

​​​​

 

​​​​

 

解该方程组，得到下面的公式:

​​​​

 

​​​​

 

把这二个公式相减，可以得到:

即：

我们也经常利用这个公式简化运算

 

基本的动量守恒演示：

先给ball类添加一个质量"属性"

package {
  import flash.display.Sprite;

  //小球 类
  public class Ball extends Sprite {

    public var radius:uint;//半径
    public var color:uint;//颜色
    public var vx:Number=0;//x轴速度
    public var vy:Number=0;//y轴速度
    public var count:uint=0;//辅助计数变量
    public var isDragged=false;//是否正在被拖动
    public var vr:Number=0;//旋转速度
    public var mass:Number = 1;//质量

    public function Ball(r:Number=50,c:uint=0xff0000) {
      this.radius=r;
      this.color=c;
      init();
    }

    private function init():void {
      graphics.beginFill(color);
      graphics.drawCircle(0,0,radius);
      graphics.endFill();
    }
  }
}

一维单轴刚体碰撞测试：

package {
  import flash.display.Sprite;
  import flash.events.Event;
  public class Billiard1 extends Sprite {
    private var ball0:Ball;
    private var ball1:Ball;
    private var bounce:Number = -0.6;
    public function Billiard1() {
      init();
    }
    
    private function init():void {
      ball0=new Ball(40);     
      addChild(ball0);
      ball1=new Ball(20,0x0000ff);      
      addChild(ball1);          
      ReStart();
    }
    
    private function ReStart():void{
      ball0.mass=2;
      ball0.x=50;
      ball0.y=stage.stageHeight/2;
      ball0.vx=5;
      ball1.mass=1;
      ball1.x=300;
      ball1.y=stage.stageHeight/2;
      ball1.vx=-5;    
      addEventListener(Event.ENTER_FRAME,EnterFrameHandler);  
    }
    
    private function EnterFrameHandler(event:Event):void {
      ball0.x+=ball0.vx;
      ball1.x+=ball1.vx;
      var dist:Number=ball1.x-ball0.x;
      
      //如果撞到了
      if (Math.abs(dist)<ball0.radius+ball1.radius) {
        var vdx:Number = ball0.vx - ball1.vx;
        var vx0Final:Number=((ball0.mass-ball1.mass)*ball0.vx + 2*ball1.mass*ball1.vx)/(ball0.mass+ball1.mass);
        var vx1Final:Number= vx0Final + vdx;
        ball0.vx=vx0Final;
        ball1.vx=vx1Final;
        
        //不加下面这二句的话，从视觉效果上看，有可能会看到二个球相互撞入对方球体内了，这样就不符合物理学"刚体"模型的定义
        ball0.x+=ball0.vx;
        ball1.x+=ball1.vx;
      }
      
      //舞台边界反弹
      if (ball0.x >=stage.stageWidth-ball0.radius || ball0.x<=ball0.radius){
        ball0.x -= ball0.vx;
        ball0.vx *= bounce;
      }     
      
      if (ball1.x >=stage.stageWidth-ball1.radius || ball1.x<=ball1.radius){
        ball1.x -= ball1.vx;
        ball1.vx *= bounce;
      }
      
      trace(ball1.vx,ball0.vx);
      
      //如果二球都停了
      if (Math.abs(ball1.vx)<=0.05 && Math.abs(ball0.vx)<=0.05){
        removeEventListener(Event.ENTER_FRAME,EnterFrameHandler); 
        ReStart();
      }
    }
  }

}

二维坐标上的刚体碰撞：

先来看这张图，红球a以Va速度运动，蓝球b以Vb速度运动，二球的连线正好与x轴平行(即：水平对心碰撞)，碰撞的过程可以理解为二球水平速度分量Vax,Vbx应用运量守恒与能力守恒的结果(y轴方向的速度不受影响！)

但很多情况下，二球的连线并非总是与坐标轴平行，比如下面这样：

思路：仍然利用坐标旋转，先将二个球反向旋转到连线水平位置，然后按常规方式处理，完事后再旋转回来。

var ballA:Ball=new Ball(80,Math.random()*0xffffff);
var ballB:Ball=new Ball(50,Math.random()*0xffffff);
var bounce:Number=-1;

ballA.x=ballA.radius+100;
ballB.x=ballA.radius+200;
ballA.y=120;
ballB.y=300;

ballA.mass=2;
ballB.mass=1;

ballA.vx = 5*(Math.random()*2-1);
ballB.vx = 5*(Math.random()*2-1);
ballA.vy = 5*(Math.random()*2-1);
ballB.vy = 5*(Math.random()*2-1);

addChild(ballA);
addChild(ballB);

addEventListener(Event.ENTER_FRAME,EnterFrameHandler);

function EnterFrameHandler(e:Event):void {
  ballA.x+=ballA.vx;
  ballA.y+=ballA.vy;
  ballB.x+=ballB.vx;
  ballB.y+=ballB.vy;
  
  //运量守恒处理开始
  var dx:Number=ballB.x-ballA.x;
  var dy:Number=ballB.y-ballA.y;
  var dist:Number=Math.sqrt(dx*dx+dy*dy);
  if (dist<(ballA.radius + ballB.radius)) {
    var angle:Number=Math.atan2(dy,dx);
    var cos:Number=Math.cos(angle);
    var sin:Number=Math.sin(angle);

    //以ballA中心为旋转中心反向旋转
    var xA:Number=0;//ballA自身为旋转中心，所以自身旋转后的相对坐标都是0
    var yA:Number=0;

    var xB:Number=dx*cos+dy*sin;
    var yB:Number=dy*cos-dx*sin;

    //先(反向)旋转二球相对(ballA的)速度
    var vxA=ballA.vx*cos+ballA.vy*sin;
    var vyA=ballA.vy*cos-ballA.vx*sin;
    var vxB=ballB.vx*cos+ballB.vy*sin;
    var vyB=ballB.vy*cos-ballB.vx*sin;

    //旋转后的vx速度处理运量守恒
    var vdx=vxA-vxB;
    var vxAFinal = ((ballA.mass - ballB.mass)*vxA + 2*ballB.mass*vxB)/(ballA.mass + ballB.mass);
    var vxBFinal=vxAFinal+vdx;

    //相对位置处理
    xA+=vxAFinal;
    xB+=vxBFinal;

    //处理完了,再旋转回去
    //先处理坐标位置
    var xAFinal:Number=xA*cos-yA*sin;
    var yAFinal:Number=yA*cos+xA*sin;
    var xBFinal:Number=xB*cos-yB*sin;
    var yBFinal:Number=yB*cos+xB*sin;

    //处理最终的位置变化
    ballB.x=ballA.x+xBFinal;
    ballB.y=ballA.y+yBFinal;
    ballA.x+=xAFinal;
    ballA.y+=yAFinal;

    //再处理速度
    ballA.vx=vxAFinal*cos-vyA*sin;
    ballA.vy=vyA*cos+vxAFinal*sin;
    ballB.vx=vxBFinal*cos-vyB*sin;
    ballB.vy=vyB*cos+vxBFinal*sin;
  }
  //<--- 运量守恒处理结束
  
  CheckBounds(ballA);
  CheckBounds(ballB);
}

//舞台边界检测
function CheckBounds(b:Ball) {
  if (b.x<b.radius) {
    b.x=b.radius;
    b.vx*=bounce;
  } else if (b.x>stage.stageWidth-b.radius) {
    b.x=stage.stageWidth-b.radius;
    b.vx*=bounce;
  }

  if (b.y<b.radius) {
    b.y=b.radius;
    b.vy*=bounce;
  } else if (b.y>stage.stageHeight-b.radius) {
    b.y=stage.stageHeight-b.radius;
    b.vy*=bounce;
  }
}

粘连问题：

反复运行上面这段动画，偶尔可能会发现二个球最终粘在一起，无法分开了，造成这种原因的情况很多，下面的示意图分析了可能的形成原因之一

解决思路：找出重叠部分，然后把二个小球同时反向移动适当距离，让二个球分开即可

先来一段测试代码：验证一下是否有效

var ballA:Ball=new Ball(80,0xff0000);
ballA.x=stage.stageWidth/2;
ballA.y=stage.stageHeight/2;
addChild(ballA);

var ballB:Ball=new Ball(60,0x00ff00);
ballB.x=stage.stageWidth/2-70;
ballB.y=stage.stageHeight/2;
addChild(ballB);

btn1.x=stage.stageWidth/2;
btn1.y=stage.stageHeight-btn1.height;
btn1.addEventListener(MouseEvent.MOUSE_DOWN,MouseDownHandler);

function MouseDownHandler(e:MouseEvent):void {
  var overlap:Number=ballA.radius+ballB.radius-Math.abs(ballA.x-ballB.x);//计算重叠部分
  trace(overlap);
  
  //计算每个球所占重叠部分中的比例
  var aRadio:Number = ballA.radius/(ballA.radius + ballB.radius);
  var bRadio:Number = ballB.radius/(ballA.radius + ballB.radius);
  
  //分离判断
  if (overlap>0){
    if (ballA.x>ballB.x){
      ballA.x += overlap*aRadio;
      ballB.x -= overlap*bRadio;
    }
    else{
      ballA.x -= overlap*aRadio;
      ballB.x += overlap*bRadio;
    }
  }
}

ballA.addEventListener(MouseEvent.MOUSE_DOWN,startDragHandler);
ballB.addEventListener(MouseEvent.MOUSE_DOWN,startDragHandler);
ballA.addEventListener(MouseEvent.MOUSE_OVER,MouseOverHandler);
ballA.addEventListener(MouseEvent.MOUSE_OUT,MouseOutHandler);
ballB.addEventListener(MouseEvent.MOUSE_OVER,MouseOverHandler);
ballB.addEventListener(MouseEvent.MOUSE_OUT,MouseOutHandler);

stage.addEventListener(MouseEvent.MOUSE_UP,stopDragHandler);

var obj:Ball;
var rect:Rectangle = new Rectangle(0,stage.stageHeight/2,stage.stageWidth,0);
function startDragHandler(e:MouseEvent):void {
  Mouse.cursor = MouseCursor.HAND;
  obj=e.currentTarget as Ball;
  obj.startDrag();
}

function stopDragHandler(e:MouseEvent):void {
  if (obj!=null) {
    obj.stopDrag(true,rect);
    obj=null;
    Mouse.cursor = MouseCursor.AUTO;
  }
}

function MouseOverHandler(e:MouseEvent):void{
  Mouse.cursor = MouseCursor.HAND;
}

function MouseOutHandler(e:MouseEvent):void{
  Mouse.cursor = MouseCursor.AUTO;
}

水平拖动小球故意让它们重叠，然后点击“分开”按钮测试一下，ok，管用了！

再回过头来解决运量守恒中的粘连问题:

只要把EnterFrameHandler中的

 //相对位置处理  

 xA+=vxAFinal;  

 xB+=vxBFinal;  

换成:

//相对位置处理(同时要防止粘连)
//xA+=vxAFinal;
//xB+=vxBFinal;
var sumRadius = ballA.radius + ballB.radius;
var overlap:Number=sumRadius-Math.abs(xA-xB);//计算重叠部分
//trace(overlap);
  
//计算每个球所占重叠部分中的比例
var aRadio:Number = ballA.radius/sumRadius;
var bRadio:Number = ballB.radius/sumRadius;
  
//分离判断
if (overlap>0){
  if (xA>xB){
    xA += overlap*aRadio;
    xB -= overlap*bRadio;
  }
  else{
    xA -= overlap*aRadio;
    xB += overlap*bRadio;
  }
}

最后老规矩：来一个群魔乱舞，把一堆球放在一块儿乱撞

package {

  import flash.display.Sprite;
  import flash.events.Event;
  import flash.geom.Point;

  public class MultiBilliard extends Sprite {

    private var balls:Array;
    private var numBalls:uint=8;
    private var bounce:Number=-1.0;

    public function MultiBilliard() {
      init();
    }

    private function init():void {
      balls = new Array();
      for (var i:uint = 0; i < numBalls; i++) {
        var radius:Number=Math.random()*40+10;
        var ball:Ball=new Ball(radius,Math.random()*0xffffff);
        ball.mass=radius;
        ball.x=i*100;
        ball.y=i*50;
        ball.vx=Math.random()*10-5;
        ball.vy=Math.random()*10-5;
        addChild(ball);
        balls.push(ball);
      }
      addEventListener(Event.ENTER_FRAME, onEnterFrame);
    }

    private function onEnterFrame(event:Event):void {
      for (var i:uint = 0; i < numBalls; i++) {
        var ball:Ball=balls[i];
        ball.x+=ball.vx;
        ball.y+=ball.vy;
        checkWalls(ball);
      }

      for (i = 0; i < numBalls - 1; i++) {
        var ballA:Ball=balls[i];
        for (var j:Number = i + 1; j < numBalls; j++) {
          var ballB:Ball=balls[j];
          checkCollision(ballA, ballB);
        }
      }
    }
    

    //舞台边界检测
    function checkWalls(b:Ball) {
      if (b.x<b.radius) {
        b.x=b.radius;
        b.vx*=bounce;
      } else if (b.x>stage.stageWidth-b.radius) {
        b.x=stage.stageWidth-b.radius;
        b.vx*=bounce;
      }
      if (b.y<b.radius) {
        b.y=b.radius;
        b.vy*=bounce;
      } else if (b.y>stage.stageHeight-b.radius) {
        b.y=stage.stageHeight-b.radius;
        b.vy*=bounce;
      }
    }

    private function rotate(x:Number, y:Number, sin:Number, cos:Number, reverse:Boolean):Point {
      var result:Point = new Point();
      if (reverse) {
        result.x=x*cos+y*sin;
        result.y=y*cos-x*sin;
      } else {
        result.x=x*cos-y*sin;
        result.y=y*cos+x*sin;
      }
      return result;
    }

    private function checkCollision(ball0:Ball, ball1:Ball):void {
      var dx:Number=ball1.x-ball0.x;
      var dy:Number=ball1.y-ball0.y;
      var dist:Number=Math.sqrt(dx*dx+dy*dy);
      if (dist<ball0.radius+ball1.radius) {
        // 计算角度和正余弦值 
        var angle:Number=Math.atan2(dy,dx);
        var sin:Number=Math.sin(angle);
        var cos:Number=Math.cos(angle);
        // 旋转 ball0 的位置 
        var pos0:Point=new Point(0,0);
        // 旋转 ball1 的速度 
        var pos1:Point=rotate(dx,dy,sin,cos,true);
        // 旋转 ball0 的速度 
        var vel0:Point=rotate(ball0.vx,ball0.vy,sin,cos,true);
        // 旋转 ball1 的速度 
        var vel1:Point=rotate(ball1.vx,ball1.vy,sin,cos,true);
        // 碰撞的作用力 
        var vxTotal:Number=vel0.x-vel1.x;
        vel0.x = ((ball0.mass - ball1.mass) * vel0.x + 2 * ball1.mass * vel1.x) / (ball0.mass + ball1.mass);
        vel1.x = vxTotal+vel0.x;
        // 更新位置 
        var absV:Number=Math.abs(vel0.x)+Math.abs(vel1.x);
        var overlap:Number = (ball0.radius + ball1.radius) - Math.abs(pos0.x - pos1.x);
        pos0.x += vel0.x/absV*overlap;
        pos1.x += vel1.x/absV*overlap;
        // 将位置旋转回来 
        var pos0F:Object=rotate(pos0.x,pos0.y,sin,cos,false);
        var pos1F:Object=rotate(pos1.x,pos1.y,sin,cos,false);
        // 将位置调整为屏幕的实际位置 
        ball1.x=ball0.x+pos1F.x;
        ball1.y=ball0.y+pos1F.y;
        ball0.x=ball0.x+pos0F.x;
        ball0.y=ball0.y+pos0F.y;
        // 将速度旋转回来 
        var vel0F:Object=rotate(vel0.x,vel0.y,sin,cos,false);
        var vel1F:Object=rotate(vel1.x,vel1.y,sin,cos,false);
        ball0.vx=vel0F.x;
        ball0.vy=vel0F.y;
        ball1.vx=vel1F.x;
        ball1.vy=vel1F.y;
      }
    }
  }
}

注：这段代码做了优化，把一些公用的部分提取出来封装成function了，同时对于粘连问题的解决，采用了更一种算法

后记：弄懂了本文中的这些玩意儿有啥用呢？让我想想，或许...公司需要开发一款桌面台球游戏时，这东西就能派上用场吧.

作者：菩提树下的杨过