Calculate velocity and direction of a ball to ball collision based on mass and bouncing coefficient

Question

I used the following code based on this

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);

but it obviously doesn't well as the formula is designed for one-dimensional collisions.

So I tried to use the below formula from this section.

But the problem is that I don't know what the angle of deflection is and how to calculate it. Also, how to take into account the bouncing coefficient in this formula?

Edit: I may have not been clear. The above code does work, although it may not be the expected behavior, as the original formula is designed for 1D collisions. The issues I'm trying therefore are:

• What is the 2D equivalent?
• How to take the bouncing coefficient into account?
• How to calculate the direction (which is expressed with v_x and v_y) of the two balls following the collision?

Answer 1

I should start by saying: I created a new answer because I feel the old one has value for its simplicity

as promised here is a much more complex physics engine, yet I still feel it's simple enough to follow (hopefully! or I just wasted my time... lol), (url: 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);

the real "meat" and the origin for this discussion is the obj.collide(obj) method.

if we dive in (I commented it this time as it is much more complex than the "last"), you'll see that this equation: , is still the only one being used in this line: this.v.tx(dn.multiply((cr * obj.m * (v2 - v1) + this.m * v1 + obj.m * v2) / sm)); now I'm sure you're still saying: "zomg wtf! that's the same single dimension equation!" but when you stop and think about it a "collision" only ever happens in a single dimension. Which is why we use vector equations to extract the applicable components and apply the collisions only to those specific parts leaving the others untouched to go on their merry way (ignoring friction and simplifying the collision to not account for dynamic energy transforming forces as described in the comments for CR). This concept obviously gets more complicated as the object complexity grows and number of scene data points increases to account for things like deformity, rotational inertia, uneven mass distribution and points of friction... but that's so far beyond the scope of this it's almost not worth mentioning..

Basically, the concepts you really need to "grasp" for this to feel intuitive to you are the basics of Vector equations (all located in the Vector prototype), how they interact with each (what it actually means to normalize, or take a dot/scalar product, eg. reading/talking to someone knowledgeable) and a basic understanding of how collisions act on properties of an object (mass, speed, etc... again, read/talk to someone knowledgeable)

I hope this helps, good luck! -ck