Collision Reactions
One way of reacting to collisions is to use a penalty-based system which basically applies a set force to a point upon contact. The problem with this is that it is very difficult to choose the force imparted. Use too strong a force and objects will become unstable, too weak and the objects will penetrate each other. Another way is to use projection collision reactions which takes the offending point and attempts to move it the shortest distance possible to move it out of the other object.
The Verlet integration would automatically handle the velocity imparted via the collision in the latter case, however note that this is not guaranteed to do so in a way that is consistent with collision physics (that is, changes in momentum are not guaranteed to be realistic). Instead of implicitly changing the velocity term, you would need to explicitly control the final velocities of the objects colliding (by changing the recorded position from the previous time step).
The two simplest methods for deciding on a new velocity are perfectly elastic collisions and inelastic collisions. A slightly more complicated strategy that offers more control would involve using the coefficient of restitution.
Read more about this topic: Verlet Integration
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