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.