我使用了以下基于这里的代码。
ballA.vx = (u1x * (m1 - m2) + 2 * m2 * u2x) / (m1 + m2);
ballA.vy = (u1y * (m1 - m2) + 2 * m2 * u2y) / (m1 + m2);
ballB.vx = (u2x * (m2 - m1) + 2 * m1 * u1x) / (m1 + m2);
ballB.vy = (u2y * (m2 - m1) + 2 * m1 * u1y) / (m1 + m2);
但是很明显，它不适用于设计为一维碰撞的公式。
因此，我尝试使用本节中的以下公式。
但问题是，我不知道偏转角度是多少，以及如何计算它。此外，在这个公式中如何考虑反弹系数？
编辑：我的表述可能不够清楚。上面的代码确实可以工作，尽管它可能不是预期的行为，因为原始公式是设计为1D碰撞的。因此，我正在尝试解决以下问题：
我应该先说：我创建了一个新答案，因为旧的答案因其简单性而具有价值。
就像我承诺的那样，这里有一个更加复杂的物理引擎，但我仍然觉得它足够简单易懂（希望如此！否则我只是在浪费时间...哈哈），（网址：http://jsbin.com/otipiv/edit#javascript,live）
function Vector(x, y) {
this.x = x;
this.y = y;
}
Vector.prototype.dot = function (v) {
return this.x * v.x + this.y * v.y;
};
Vector.prototype.length = function() {
return Math.sqrt(this.x * this.x + this.y * this.y);
};
Vector.prototype.normalize = function() {
var s = 1 / this.length();
this.x *= s;
this.y *= s;
return this;
};
Vector.prototype.multiply = function(s) {
return new Vector(this.x * s, this.y * s);
};
Vector.prototype.tx = function(v) {
this.x += v.x;
this.y += v.y;
return this;
};
function BallObject(elasticity, vx, vy) {
this.v = new Vector(vx || 0, vy || 0); // velocity: m/s^2
this.m = 10; // mass: kg
this.r = 15; // radius of obj
this.p = new Vector(0, 0); // position
this.cr = elasticity; // elasticity
}
BallObject.prototype.draw = function(ctx) {
ctx.beginPath();
ctx.arc(this.p.x, this.p.y, this.r, 0, 2 * Math.PI);
ctx.closePath();
ctx.fill();
ctx.stroke();
};
BallObject.prototype.update = function(g, dt, ppm) {
this.v.y += g * dt;
this.p.x += this.v.x * dt * ppm;
this.p.y += this.v.y * dt * ppm;
};
BallObject.prototype.collide = function(obj) {
var dt, mT, v1, v2, cr, sm,
dn = new Vector(this.p.x - obj.p.x, this.p.y - obj.p.y),
sr = this.r + obj.r, // sum of radii
dx = dn.length(); // pre-normalized magnitude
if (dx > sr) {
return; // no collision
}
// sum the masses, normalize the collision vector and get its tangential
sm = this.m + obj.m;
dn.normalize();
dt = new Vector(dn.y, -dn.x);
// avoid double collisions by "un-deforming" balls (larger mass == less tx)
// this is susceptible to rounding errors, "jiggle" behavior and anti-gravity
// suspension of the object get into a strange state
mT = dn.multiply(this.r + obj.r - dx);
this.p.tx(mT.multiply(obj.m / sm));
obj.p.tx(mT.multiply(-this.m / sm));
// this interaction is strange, as the CR describes more than just
// the ball's bounce properties, it describes the level of conservation
// observed in a collision and to be "true" needs to describe, rigidity,
// elasticity, level of energy lost to deformation or adhesion, and crazy
// values (such as cr > 1 or cr < 0) for stange edge cases obviously not
// handled here (see: http://en.wikipedia.org/wiki/Coefficient_of_restitution)
// for now assume the ball with the least amount of elasticity describes the
// collision as a whole:
cr = Math.min(this.cr, obj.cr);
// cache the magnitude of the applicable component of the relevant velocity
v1 = dn.multiply(this.v.dot(dn)).length();
v2 = dn.multiply(obj.v.dot(dn)).length();
// maintain the unapplicatble component of the relevant velocity
// then apply the formula for inelastic collisions
this.v = dt.multiply(this.v.dot(dt));
this.v.tx(dn.multiply((cr * obj.m * (v2 - v1) + this.m * v1 + obj.m * v2) / sm));
// do this once for each object, since we are assuming collide will be called
// only once per "frame" and its also more effiecient for calculation cacheing
// purposes
obj.v = dt.multiply(obj.v.dot(dt));
obj.v.tx(dn.multiply((cr * this.m * (v1 - v2) + obj.m * v2 + this.m * v1) / sm));
};
function FloorObject(floor) {
var py;
this.v = new Vector(0, 0);
this.m = 5.9722 * Math.pow(10, 24);
this.r = 10000000;
this.p = new Vector(0, py = this.r + floor);
this.update = function() {
this.v.x = 0;
this.v.y = 0;
this.p.x = 0;
this.p.y = py;
};
// custom to minimize unnecessary filling:
this.draw = function(ctx) {
var c = ctx.canvas, s = ctx.scale;
ctx.fillRect(c.width / -2 / s, floor, ctx.canvas.width / s, (ctx.canvas.height / s) - floor);
};
}
FloorObject.prototype = new BallObject(1);
function createCanvasWithControls(objs) {
var addBall = function() { objs.unshift(new BallObject(els.value / 100, (Math.random() * 10) - 5, -20)); },
d = document,
c = d.createElement('canvas'),
b = d.createElement('button'),
els = d.createElement('input'),
clr = d.createElement('input'),
cnt = d.createElement('input'),
clrl = d.createElement('label'),
cntl = d.createElement('label');
b.innerHTML = 'add ball with elasticity: <span>0.70</span>';
b.onclick = addBall;
els.type = 'range';
els.min = 0;
els.max = 100;
els.step = 1;
els.value = 70;
els.style.display = 'block';
els.onchange = function() {
b.getElementsByTagName('span')[0].innerHTML = (this.value / 100).toFixed(2);
};
clr.type = cnt.type = 'checkbox';
clr.checked = cnt.checked = true;
clrl.style.display = cntl.style.display = 'block';
clrl.appendChild(clr);
clrl.appendChild(d.createTextNode('clear each frame'));
cntl.appendChild(cnt);
cntl.appendChild(d.createTextNode('continuous shower!'));
c.style.border = 'solid 1px #3369ff';
c.style.display = 'block';
c.width = 700;
c.height = 550;
c.shouldClear = function() { return clr.checked; };
d.body.appendChild(c);
d.body.appendChild(els);
d.body.appendChild(b);
d.body.appendChild(clrl);
d.body.appendChild(cntl);
setInterval(function() {
if (cnt.checked) {
addBall();
}
}, 333);
return c;
}
// start:
var objs = [],
c = createCanvasWithControls(objs),
ctx = c.getContext('2d'),
fps = 30, // target frames per second
ppm = 20, // pixels per meter
g = 9.8, // m/s^2 - acceleration due to gravity
t = new Date().getTime();
// add the floor:
objs.push(new FloorObject(c.height - 10));
// as expando so its accessible in draw [this overides .scale(x,y)]
ctx.scale = 0.5;
ctx.fillStyle = 'rgb(100,200,255)';
ctx.strokeStyle = 'rgb(33,69,233)';
ctx.transform(ctx.scale, 0, 0, ctx.scale, c.width / 2, c.height / 2);
setInterval(function() {
var i, j,
nw = c.width / ctx.scale,
nh = c.height / ctx.scale,
nt = new Date().getTime(),
dt = (nt - t) / 1000;
if (c.shouldClear()) {
ctx.clearRect(nw / -2, nh / -2, nw, nh);
}
for (i = 0; i < objs.length; i++) {
// if a ball > viewport width away from center remove it
while (objs[i].p.x < -nw || objs[i].p.x > nw) {
objs.splice(i, 1);
}
objs[i].update(g, dt, ppm, objs, i);
for (j = i + 1; j < objs.length; j++) {
objs[j].collide(objs[i]);
}
objs[i].draw(ctx);
}
t = nt;
}, 1000 / fps);
obj.collide(obj) 方法。
，仍然是这行代码中唯一使用的方程式：this.v.tx(dn.multiply((cr * obj.m * (v2 - v1) + this.m * v1 + obj.m * v2) / sm));。 现在我敢肯定你还在说：“zomg wtf！这是同一个单维度方程！”但当你停下来想一想，“碰撞”只发生在一个维度上。这就是为什么我们使用向量方程式提取适用的组件，并仅将碰撞应用于这些特定部分，使其他部分保持不变而朝着自己的方式前进（忽略摩擦并简化碰撞以不考虑动态能量转换力，如 CR 的评论所述）。 当物体复杂性增加且场景数据点的数量增加以满足变形、旋转惯量、不均匀质量分布和摩擦点等因素时，该概念显然变得更加复杂...但这已经远远超出了范围，几乎不值得一提。这里是一个非弹性碰撞方程式的演示，专门为您定制：
function BallObject(elasticity) {
this.v = { x: 1, y: 20 }; // velocity: m/s^2
this.m = 10; // mass: kg
this.p = { x: 40, y: 0}; // position
this.r = 15; // radius of obj
this.cr = elasticity; // elasticity
}
function draw(obj) {
ctx.beginPath();
ctx.arc(obj.p.x, obj.p.y, obj.r, 0, 2 * Math.PI);
ctx.closePath();
ctx.stroke();
ctx.fill();
}
function collide(obj) {
obj.v.y = (obj.cr * floor.m * -obj.v.y + obj.m * obj.v.y) / (obj.m + floor.m);
}
function update(obj, dt) {
// over-simplified collision detection
// only consider the floor for simplicity
if ((obj.p.y + obj.r) > c.height) {
obj.p.y = c.height - obj.r;
collide(obj);
}
obj.v.y += g * dt;
obj.p.x += obj.v.x * dt * ppm;
obj.p.y += obj.v.y * dt * ppm;
}
var d = document,
c = d.createElement('canvas'),
b = d.createElement('button'),
els = d.createElement('input'),
clr = d.createElement('input'),
clrl = d.createElement('label'),
ctx = c.getContext('2d'),
fps = 30, // target frames per second
ppm = 20, // pixels per meter
g = 9.8, // m/s^2 - acceleration due to gravity
objs = [],
floor = {
v: { x: 0, y: 0 }, // floor is immobile
m: 5.9722 * Math.pow(10, 24) // mass of earth (probably could be smaller)
},
t = new Date().getTime();
b.innerHTML = 'add ball with elasticity: <span>0.70</span>';
b.onclick = function() { objs.push(new BallObject(els.value / 100)); };
els.type = 'range';
els.min = 0;
els.max = 100;
els.step = 1;
els.value = 70;
els.style.display = 'block';
els.onchange = function() {
b.getElementsByTagName('span')[0].innerHTML = (this.value / 100).toFixed(2);
};
clr.type = 'checkbox';
clr.checked = true;
clrl.appendChild(clr);
clrl.appendChild(d.createTextNode('clear each frame'));
c.style.border = 'solid 1px #3369ff';
c.style.borderRadius = '10px';
c.style.display = 'block';
c.width = 400;
c.height = 400;
ctx.fillStyle = 'rgb(100,200,255)';
ctx.strokeStyle = 'rgb(33,69,233)';
d.body.appendChild(c);
d.body.appendChild(els);
d.body.appendChild(b);
d.body.appendChild(clrl);
setInterval(function() {
var nt = new Date().getTime(),
dt = (nt - t) / 1000;
if (clr.checked) {
ctx.clearRect(0, 0, c.width, c.height);
}
for (var i = 0; i < objs.length; i++) {
update(objs[i], dt);
draw(objs[i]);
}
t = nt;
}, 1000 / fps);
由于您的“地板”不会移动，因此您只需要考虑对球的y速度的影响。请注意，在此处存在许多捷径和疏忽，因此这是一个非常原始的物理引擎，并且主要用于说明这个方程式...希望这可以帮到您-ck我强烈建议您熟悉动量中心参考系。这使得碰撞的理解变得容易得多。（如果没有这种理解，您只是在操作晦涩的方程式，而不知道为什么事情会出错。）
无论如何，要确定角度，您可以使用冲击参数，基本上是一个球离中心有多远就撞到了另一个球。两个球在相反的方向（在动量中心参考系中）相向而行，它们之间垂直于那些速度的中心距离是冲击参数h。然后偏转角度为2 acos(h/(r1+r2))。
一旦您完美地掌握了这个，您可以开始担心非弹性碰撞和恢复系数。