2d Elastic Collision with Circles - c++

Question

I've seen there is a lot of posts about this already but I can't find one that relates to what I want to do,
I used the formula from here:
https://www.vobarian.com/collisions/2dcollisions2.pdf
As well as this one:
https://www.plasmaphysics.org.uk/programs/coll2d_cpp.htm
I think they area basically the same thing, now my problem is one of my circles is always static, and what I want is when the other circle hits it straight on, I want it to bounce back with the same speed, but these formulas have the circle stop still, presumably as it would pass it's energy to the other circle which would then move away.
I tried doing things like bounce = vel.x pre collision - vel.y post collision and add or subtract that to vel.x post collision and it kinda works but not really, the angles are wrong and depending on which direction the ball is coming from it may bounce up instead of down, left instead of right,
would probably require a lot of if/else statements to get to work at all.
Can someone suggest something?
here's the code for the function :
void Collision2(sf::CircleShape* b1, sf::CircleShape* b2, sf::Vector2f vel1,sf::Vector2f& vel2) {
//vel1 is 0,0 but i might want to use it later
//mass
float m1 = 10;
float m2 = 10;
//normal vector
sf::Vector2f nVec((b2->getPosition().x - b1->getPosition().x), (b2->getPosition().y - b1->getPosition().y));
//unit vector
sf::Vector2f uNVec(nVec / sqrt((nVec.x * nVec.x) + (nVec.y * nVec.y)));
//unit tangent vec
sf::Vector2f uTVec(-uNVec.y, uNVec.x);
float v1n = (uNVec.x * vel1.x) + (uNVec.y * vel1.y);
float v2n = (uNVec.x * vel2.x) + (uNVec.y * vel2.y);
float v1t = uTVec.x * vel1.x + uTVec.y * vel2.y;
float v2t = (uTVec.x * vel2.x) + (uTVec.y * vel2.y);
//v1t and v1n after collision
float v1tN = v1t;
float v2tN = v2t;
float v1nN = (v1n * (m1 - m2) + (2 * m2) * v2n) / (m1 + m2);
float v2nN = (v2n * (m2 - m1) + (2 * m1) * v1n) / (m1 + m2);
//new velocities
sf::Vector2f vel1N(v1nN*uNVec);
sf::Vector2f vel1tN(v1tN * uTVec);
sf::Vector2f vel2N(v2nN * uNVec);
sf::Vector2f vel2tN(v2tN * uTVec);
vel1 = (vel1N + vel1tN);
vel2 = (vel2N + vel2tN);
}

Answer 1

Physics part
The sources you added illustrate the physics behind it very well. when the two balls collide they transfer momentum between them. In an elastic collision this transfer keeps the energy of the system, the same.
We can think of the collision in terms of inertia and momentum, rather than starting from velocity. The kinetic energy of a body is generally p^2/(2m), so if we transfer dp from the moving body then we will have change in energy: dE = -pdp/m + dp^2/(2m) + dp^2/(2M) = 0. Here m is the moving and M is the stationary mass. Rearranging gives pdp/m = dp^2*(1/(2m) + 1/(2M)). We can consider m = M yielding p = dp (i.e. All moment is transferred (Note: this is a simplistic view, only dealing with head on collisions)). In the limit where the stationary object is massive however (M >> m) the result will be dp = 2p, simply bouncing off.
Programming
You can achieve the results by setting M to the maximum allowed float value (if I recall 1/inf == NaN in the IEEE standard so that doesn't work unfortunately). Alternatively you can do the collision within the circle by creating custom classes like:
class Circle : public sf::CircleShape{
public:
virtual collide (Circle*);
}
class StaticCircle : public Circle{
public:
collide (Circle*) override;
}
in the second one you can omit any terms where you divide by the mass of the circle, as it is in essence infinite.