Flash/Flex学习笔记(38)：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.