Managing ball-to-ball collisions in a simulation involving multiple interacting objects necessitates an understanding of several fundamental principles, such as collision detection, the application of physical laws (including momentum conservation and restitution), and the efficient resolution of these interactions.

When utilizing the Verlet integration method, a numerical approach for simulating object motion, the process of managing ball-to-ball collisions can be executed in a series of steps:

1. Fundamentals of Verlet Integration Position Update: The Verlet algorithm modifies an object's position by referencing its previous and current positions:

r(t + Δt) = 2r(t) - r(t - Δt) + a(t)Δt²

Where:

r(t) represents the position of the ball at time t, a(t) denotes the acceleration (resulting from gravity, friction, etc.) at time t, Δt signifies the time increment. Velocity Calculation: The velocity can be estimated by the change in position over the time interval:

v(t) = (r(t) - r(t - Δt)) / Δt

2. Collision Detection To ascertain whether two balls collide, it is essential to evaluate if the distance between their centers is less than the sum of their radii:

∥r₁ - r₂∥ < r₁ + r₂

Where:

r₁ and r₂ indicate the positions of balls 1 and 2, r₁ and r₂ represent the radii of the respective balls. If this condition holds true, a collision is confirmed.

3. Collision Response (Elastic Collision) Upon the occurrence of a collision, the velocities of the balls are modified in accordance with the principles of momentum conservation and the coefficient of restitution. The primary steps include:

Relative Velocity: The relative velocity between the balls along the line of contact is expressed as: v_rel = v₁ - v₂

Normal Vector: The unit normal vector n̂ directed from ball 2 to ball 1 is defined as: n̂ = (r₁ - r₂) / ∥r₁ - r₂∥

Impulse Calculation.