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.
