function gravityForce(ball) {

    /* We simulate a kind of gravity towards the target point of each ball. This force will make the ball to keep pushing to reach that point. */

    const coulomb = .00001;
    const minForce = 0.01;

    var d = Math.sqrt(Math.pow(ball.targetPosition.x - ball.currentPosition.x, 2) + Math.pow(ball.targetPosition.y - ball.currentPosition.y, 2));  // ... we calculate the distance ...

    var force = coulomb * d * d / ball.currentMass;  // In this case the mass of the ball affects the gravity force that it receives, that gives priority to target position to bigger balls. 

    if (force < minForce) { // We need this attraction force to overcome the friction we imposed to the system.
        force = minForce;
    }

    var pos1 = {
        x: ball.currentPosition.x,
        y: ball.currentPosition.y
    };

    var pos2 = {
        x: ball.targetPosition.x,
        y: ball.targetPosition.y
    };

    var posDiff = {             // Next we need the vector resulting from the 2 positions.
        x: pos2.x - pos1.x,
        y: pos2.y - pos1.y
    };

    var unitVector = {          // To find the unit vector, we divide each component by the magnitude of the vector.
        x: posDiff.x / d,
        y: posDiff.y / d
    };

    var forceVector = {
        x: unitVector.x * force,
        y: unitVector.y * force
    };

    /* We add the force vector to the speed vector */

    ball.currentSpeed.x = ball.currentSpeed.x + forceVector.x;
    ball.currentSpeed.y = ball.currentSpeed.y + forceVector.y;

}

function repulsionForce(currentBall) {

    /* We generate a repulsion force between balls, that prevents them to be collisioning so often. */

    const coulomb = 2;

    var ball1 = balls[currentBall];

    for (var i = 0; i < balls.length; i++) {  // The force to be applied is considering all other balls...

        if (i !== currentBall) {  // ... except for the current one. 

            var ball2 = balls[i];   // So, for each ball...

            var d = Math.sqrt(Math.pow(ball2.currentPosition.x - ball1.currentPosition.x, 2) + Math.pow(ball2.currentPosition.y - ball1.currentPosition.y, 2));  // ... we calculate the distance ...

            var force = coulomb * ball2.currentMass / (d * d);  // ... and with it the repulsion force.

            /* We need to put a hard limit to this force, in order to to eject very little balls to the infinite and beyond. */

            if (force > 1) {
                force = 1;
            }

            var pos1 = {
                x: ball1.currentPosition.x,
                y: ball1.currentPosition.y
            };

            var pos2 = {
                x: ball2.currentPosition.x,
                y: ball2.currentPosition.y
            };

            var posDiff = {             // Next we need the vector resulting from the 2 positions.
                x: pos2.x - pos1.x,
                y: pos2.y - pos1.y
            };

            var unitVector = {          // To find the unit vector, we divide each component by the magnitude of the vector.
                x: posDiff.x / d,
                y: posDiff.y / d
            };

            var forceVector = {
                x: unitVector.x * force,
                y: unitVector.y * force
            };

            /* We substract the force vector to the speed vector of the current ball */

            ball1.currentSpeed.x = ball1.currentSpeed.x - forceVector.x;
            ball1.currentSpeed.y = ball1.currentSpeed.y - forceVector.y;

        }

    }

}



function colliding(ball1, ball2)

{
    /* This function detects weather 2 balls collide with each other. */

    var r1 = ball1.currentRadius;
    var r2 = ball2.currentRadius;

    var distance = Math.sqrt(Math.pow(ball2.currentPosition.x - ball1.currentPosition.x, 2) + Math.pow(ball2.currentPosition.y - ball1.currentPosition.y, 2));

    if (distance > (r1 + r2)) {
        // No solutions, the circles are too far apart.  
        return false;
    }
    else if (distance <= r1 + r2) {
        // One circle contains the other.
        return true;
    }
    else if ((distance === 0) && (r1 === r2)) {
        // the circles coincide.
        return true;
    }
    else return true;
}


function isInside(x, y) {

    /* This function detects weather the point x,y is inside any of the balls. */

    for (var i = 0; i < balls.length; i++) {
        var ball = balls[i];
        var distance = Math.sqrt(Math.pow(ball.currentPosition.x - x, 2) + Math.pow(ball.currentPosition.y - y, 2));

        if (distance < ball.currentRadius) {
            return i;
        }
    }
    return -1;

}


function isInsideBall(ballIndex, x, y) {

    /* This function detects weather the point x,y is inside one particular balls. */

   
    var ball = balls[ballIndex];
    var distance = Math.sqrt(Math.pow(ball.currentPosition.x - x, 2) + Math.pow(ball.currentPosition.y - y, 2));

    if (distance < ball.currentRadius) {
        return true;
    }
   
    return false;

}

function distance(x1, y1, x2, y2) {
    return Math.sqrt(Math.pow(x2 - x1, 2) + Math.pow(y2 - y1, 2));
}


function resolveCollision(ball1, ball2) {

    /* This function changes speed and position of balls that are in collision */

    var collisionision_angle = Math.atan2((ball2.currentPosition.y - ball1.currentPosition.y), (ball2.currentPosition.x - ball1.currentPosition.x));

    var speed1 = Math.sqrt(ball1.currentSpeed.x * ball1.currentSpeed.x + ball1.currentSpeed.y * ball1.currentSpeed.y);  // Magnitude of Speed Vector for ball 1
    var speed2 = Math.sqrt(ball2.currentSpeed.x * ball2.currentSpeed.x + ball2.currentSpeed.y * ball2.currentSpeed.y);  // Magnitude of Speed Vector for ball 2

    var direction_1 = Math.atan2(ball1.currentSpeed.y, ball1.currentSpeed.x);
    var direction_2 = Math.atan2(ball2.currentSpeed.y, ball2.currentSpeed.x);

    var new_xspeed_1 = speed1 * Math.cos(direction_1 - collisionision_angle);
    var new_yspeed_1 = speed1 * Math.sin(direction_1 - collisionision_angle);
    var new_xspeed_2 = speed2 * Math.cos(direction_2 - collisionision_angle);
    var new_yspeed_2 = speed2 * Math.sin(direction_2 - collisionision_angle);

    var final_xspeed_1 = ((ball1.currentMass - ball2.currentMass) * new_xspeed_1 + (ball2.currentMass + ball2.currentMass) * new_xspeed_2) / (ball1.currentMass + ball2.currentMass);
    var final_xspeed_2 = ((ball1.currentMass + ball1.currentMass) * new_xspeed_1 + (ball2.currentMass - ball1.currentMass) * new_xspeed_2) / (ball1.currentMass + ball2.currentMass);
    var final_yspeed_1 = new_yspeed_1;
    var final_yspeed_2 = new_yspeed_2;

    var cosAngle = Math.cos(collisionision_angle);
    var sinAngle = Math.sin(collisionision_angle);


    ball1.currentSpeed.x = cosAngle * final_xspeed_1 - sinAngle * final_yspeed_1;
    ball1.currentSpeed.y = sinAngle * final_xspeed_1 + cosAngle * final_yspeed_1;

    ball2.currentSpeed.x = cosAngle * final_xspeed_2 - sinAngle * final_yspeed_2;
    ball2.currentSpeed.y = sinAngle * final_xspeed_2 + cosAngle * final_yspeed_2;





    var pos1 = {
        x: ball1.currentPosition.x,
        y: ball1.currentPosition.y
    };

    var pos2 = {
        x: ball2.currentPosition.x,
        y: ball2.currentPosition.y
    };

    // get the mtd
    var posDiff = {
        x: pos1.x - pos2.x,
        y: pos1.y - pos2.y
    };

    var d = Math.sqrt(Math.pow(ball2.currentPosition.x - ball1.currentPosition.x, 2) + Math.pow(ball2.currentPosition.y - ball1.currentPosition.y, 2));

    // minimum translation distance to push balls apart after intersecting
    var scalar = (((ball1.currentRadius + ball2.currentRadius) - d) / d);

    var minTD = {
        x: posDiff.x * scalar,
        y: posDiff.y * scalar
    };

    // resolve intersection --
    // computing inverse mass quantities
    var im1 = 1 / ball1.currentMass;
    var im2 = 1 / ball2.currentMass;

    // push-pull them apart based off their mass

    pos1.x = pos1.x + minTD.x * (im1 / (im1 + im2));
    pos1.y = pos1.y + minTD.y * (im1 / (im1 + im2));
    pos2.x = pos2.x - minTD.x * (im2 / (im1 + im2));
    pos2.y = pos2.y - minTD.y * (im2 / (im1 + im2));

    ball1.currentPosition = pos1;
    ball2.currentPosition = pos2;
}