A trajectory is just "the path that a moving object follows through space as a function of time". Therefore, there are tons of different possible trajectories, depending on the situation you set up. Anytime you set up an equation to describe the motion of a particle/object/whatever, you are describing a trajectory.
In a similar vein, a collision does not have a trajectory...but all of the objects involved in the collision do. There are tons of interesting physics questions that require you to figure out the trajectory of an object before and after a collision, using a combination of momentum conservation and Newtonian equations.