Time interval between two objects passing an intersection points

In linear motions, a collision occurs between two objects if
$$p_1 + t_1v_1 = p_2 + t_2v_2$$
where $p_1,p_2$ are two arbitrary position vectors and $t_1,t_2$ are the time durations by which objects will cross the collision point. By taking the cross product with the encounter velocity $U = v_1 – v_2$ on both side, we will get $(p_1-p_2) times U = (t_1-t_2)(v_1times v_2)$. Thus, it can be said that for any $Delta t$, collision would happen if
$$Delta t leq frac{d_{min}U}{mid v_1 times v_2 mid }$$

Now, I want to find the minimum $Delta t$ for non-linear motions. Assuming the trajectory equations ($f(x)$ is second degree polynomial) are given, does it make sense to virtually linearize the motions and adopt the same approach? That means the length of each trajectory arch is the same as linear trajectory to collision point ($vt = intsqrt{1+f'(x)} dx$).

What is the best approach to find $Delta t$ threshold so as to predict a collision in such a non-linear motion system? And can it be modeled for motions with non-vectorized velocity and position where the position of objects and their intersection point is given along with the magnitude of velocities?