With the help of the Stack Overflow community I've written a pretty basic-but fun physics simulator.

You click and drag the mouse to launch a ball. It will bounce around and eventually stop on the "floor".

My next big feature I want to add in is ball to ball collision. The ball's movement is broken up into a x and y speed vector. I have gravity (small reduction of the y vector each step), I have friction (small reduction of both vectors each collision with a wall). The balls honestly move around in a surprisingly realistic way.

I guess my question has two parts:

1. What is the best method to detect ball to ball collision?
   Do I just have an O(n^2) loop that iterates over each ball and checks every other ball to see if it's radius overlaps?
2. What equations do I use to handle the ball to ball collisions? Physics 101
   How does it effect the two balls speed x/y vectors? What is the resulting direction the two balls head off in? How do I apply this to each ball?

Handling the collision detection of the "walls" and the resulting vector changes were easy but I see more complications with ball-ball collisions. With walls I simply had to take the negative of the appropriate x or y vector and off it would go in the correct direction. With balls I don't think it is that way.

Some quick clarifications: for simplicity I'm ok with a perfectly elastic collision for now, also all my balls have the same mass right now, but I might change that in the future.

In case anyone is interested in playing with the simulator I have made so far, I've uploaded the source here (EDIT: Check the updated source below).

Edit: Resources I have found useful

2d Ball physics with vectors: 2-Dimensional Collisions Without Trigonometry.pdf
2d Ball collision detection example: Adding Collision Detection

Success!

I have the ball collision detection and response working great!

Relevant code:

Collision Detection:

for (int i = 0; i < ballCount; i++)  
{  
    for (int j = i + 1; j < ballCount; j++)  
    {  
        if (balls[i].colliding(balls[j]))  
        {
            balls[i].resolveCollision(balls[j]);
        }
    }
}

This will check for collisions between every ball but skip redundant checks (if you have to check if ball 1 collides with ball 2 then you don't need to check if ball 2 collides with ball 1. Also, it skips checking for collisions with itself).

Then, in my ball class I have my colliding() and resolveCollision() methods:

public boolean colliding(Ball ball)
{
    float xd = position.getX() - ball.position.getX();
    float yd = position.getY() - ball.position.getY();

    float sumRadius = getRadius() + ball.getRadius();
    float sqrRadius = sumRadius * sumRadius;

    float distSqr = (xd * xd) + (yd * yd);

    if (distSqr <= sqrRadius)
    {
        return true;
    }

    return false;
}

public void resolveCollision(Ball ball)
{
    // get the mtd
    Vector2d delta = (position.subtract(ball.position));
    float d = delta.getLength();
    // minimum translation distance to push balls apart after intersecting
    Vector2d mtd = delta.multiply(((getRadius() + ball.getRadius())-d)/d); 


    // resolve intersection --
    // inverse mass quantities
    float im1 = 1 / getMass(); 
    float im2 = 1 / ball.getMass();

    // push-pull them apart based off their mass
    position = position.add(mtd.multiply(im1 / (im1 + im2)));
    ball.position = ball.position.subtract(mtd.multiply(im2 / (im1 + im2)));

    // impact speed
    Vector2d v = (this.velocity.subtract(ball.velocity));
    float vn = v.dot(mtd.normalize());

    // sphere intersecting but moving away from each other already
    if (vn > 0.0f) return;

    // collision impulse
    float i = (-(1.0f + Constants.restitution) * vn) / (im1 + im2);
    Vector2d impulse = mtd.multiply(i);

    // change in momentum
    this.velocity = this.velocity.add(impulse.multiply(im1));
    ball.velocity = ball.velocity.subtract(impulse.multiply(im2));

}

Source Code: Complete source for ball to ball collider.
Binary: Compiled binary in case you just want to try bouncing some balls around.

If anyone has some suggestions for how to improve this basic physics simulator let me know! One thing I have yet to add is angular momentum so the balls will roll more realistically. Any other suggestions? Leave a comment!