simple 2d collision problem - c++

Question

I want to find when a collision between a static and a moving ball occurs, but the algorithm I came up with, sometimes doesn't detect a collision and the moving ball goes through the static one. The moving ball is affected by gravity and the static one is not.
Here's my collision detection code:
GLfloat whenSpheresCollide(const sphere2d &firstSphere, const sphere2d &secondSphere)
{
Vector2f relativePosition = subtractVectors(firstSphere.vPosition, secondSphere.vPosition);
Vector2f relativeVelocity = subtractVectors(firstSphere.vVelocity, secondSphere.vVelocity);
GLfloat radiusSum = firstSphere.radius + secondSphere.radius;
//We'll find the time when objects collide if a collision takes place
//r(t) = P[0] + t * V[0]
//
//d^2(t) = P[0]^2 + 2 * t * P[0] * V[0] + t^2 * V[0]^2
//
//d^2(t) = V[0]^2 * t^2 + 2t( P[0] . V[0] ) + P[0]^2
//
//d(t) = R
//
//d(t)^2 = R^2
//
//V[0]^2 * t^2 + 2t( P[0] . V[0] ) + P[0]^2 - R^2 = 0
//
//delta = ( P[0] . V[0] )^2 - V[0]^2 * (P[0]^2 - R^2)
//
// We are interested in the lowest t:
//
//t = ( -( P[0] . V[0] ) - sqrt(delta) ) / V[0]^2
//
GLfloat equationDelta = squaref( dotProduct(relativePosition, relativeVelocity) ) - squarev( relativeVelocity ) * ( squarev( relativePosition ) - squaref(radiusSum) );
if (equationDelta >= 0.0f)
{
GLfloat collisionTime = ( - dotProduct(relativePosition, relativeVelocity) - sqrtf(equationDelta) ) / squarev(relativeVelocity);
if (collisionTime >= 0.0f && collisionTime <= 1.0f / GAME_FPS)
{
return collisionTime;
}
}
return -1.0f;
}
And here is the updating function that calls collision detection:
void GamePhysicsManager::updateBallPhysics()
{
//
//Update velocity
vVelocity->y -= constG / GAME_FPS; //v = a * t = g * 1 sec / (updates per second)
shouldApplyForcesToBall = TRUE;
vPosition->x += vVelocity->x / GAME_FPS;
vPosition->y += vVelocity->y / GAME_FPS;
if ( distanceBetweenVectors( *pBall->getPositionVector(), *pBasket->getPositionVector() ) <= pBasket->getRadius() + vectorLength(*vVelocity) / GAME_FPS )
{
//Ball sphere
sphere2d ballSphere;
ballSphere.radius = pBall->getRadius();
ballSphere.mass = 1.0f;
ballSphere.vPosition = *( pBall->getPositionVector() );
ballSphere.vVelocity = *( pBall->getVelocityVector() );
sphere2d ringSphereRight;
ringSphereRight.radius = 0.05f;
ringSphereRight.mass = -1.0f;
ringSphereRight.vPosition = *( pBasket->getPositionVector() );
//ringSphereRight.vPosition.x += pBasket->getRadius();
ringSphereRight.vPosition.x += (pBasket->getRadius() - ringSphereRight.radius);
ringSphereRight.vVelocity = zeroVector();
GLfloat collisionTime = whenSpheresCollide(ballSphere, ringSphereRight);
if ( collisionTime >= 0.0f )
{
DebugLog("collision");
respondToCollision(&ballSphere, &ringSphereRight, collisionTime, pBall->getRestitution() * 0.75f );
}
//
//Implement selection of the results that are first to collide collision
vVelocity->x = ballSphere.vVelocity.x;
vVelocity->y = ballSphere.vVelocity.y;
vPosition->x = ballSphere.vPosition.x;
vPosition->y = ballSphere.vPosition.y;
}
Why isn't the collision being detected in 100% of cases? It's being detected only in 70% of cases.
Thanks.
UPDATE: Problem seems to be solved when I change FPS from 30 to 10. How does FPS affect my collision detection?

Answer 1

delta = ( P[0] . V[0] )^2 - V[0]^2 * (P[0]^2 - R^2)
Shouldn't that be delta = b2 - 4 ac?
[Edit] Oh I see, you factored the 4 out. In that case, are you sure you're considering both solutions for t?
t = ( -( P[0] . V[0] ) - sqrt(delta) ) / V[0]^2
and
t = ( -( P[0] . V[0] ) + sqrt(delta) ) / V[0]^2

Answer 2

How large are the sphere's and how fast are they moving? Can a sphere "jump" over the second one during a frame (i.e., is it's velocity vector longer than it's width?).
Along those lines, what happens if you remove the upper limit here:
if (collisionTime >= 0.0f && collisionTime <= 1.0f / GAME_FPS)
{
return collisionTime;
}
If the sphere was moving too fast, maybe your algorithm is detecting a collision that happened more than one frame ago .. (?)