Let us presume that we have a scenario with ballistic objects. As an example, one object $m_1$ is launched at time $t_0$ from Earth Location $a$ towards Earth Location $b$, both of which are in the same hemisphere, positioned so that a minimum time trajectory must go over one polar region, and are between 60° and 120° apart. Then at some time $t_1$ between the first launch and its impact at $t_2$, a ballistic object $m_2$ is launched from $b$ toward the first object's launch site $a$, to impact at $t_3$.
Both ballistic objects are on flight paths with approximately the same eccentricity, chosen to minimise flight time as far as practical.
While the objects will both pass roughly over the pole, the trajectories are not superimposed since the Earth is rotating, and at closest approach the objects are a considerable distance apart.
However, while launched from $a$ in such a manner that it should not have to use it to reach $b$, $m_1$ has an on-board capability to generate delta-V, and if it has sufficient delta-V, it has the capability of targeting and manoeuvring to generate a collision with $m_2$.
Question: How can I calculate the minimum amount of delta-V required to alter the trajectory of $m_1$ starting at $t_1$ so that it will impact $m_2$ at some time between $m_2$'s launch at $t_1$ and $m_1$'s impact at $m_2$'s launch site $b$ at $t_2$ or $m_2$'s impact at $m_1$'s launch site $a$ at $t_3$?
$m_2$ should be considered to have no way of having its trajectory altered once launched.
My thinking so far:
Obviously, if $t_1$=$t_2$, then $m_1$'s delta-V required will be minimal. I am more concerned with solutions in the range $t_1 = 0$ to $t_1 = t_0 +(t_2-t_0)/2$, i.e. $m_2$'s launch at $t_1$ occurring with $m_1$ half-way along its trajectory.
Since this is a missile counter-battery question, this is not a zero-velocity rendezvous, but should be an impact that takes place at as high a closing velocity as practical, effectively meaning that no deceleration or course-matching by $m_1$ to meet $m_2$ is necessary.
I have access to AGI's free STK 11.2 software package, but I have none of the paid add-ons that would make this question unnecessary, since I am not only short of funds, but I am also not American, and hence prohibited from receiving the most relevant of AGI's packages even as a trial due to US export restrictions, and I have also exhausted the use of trial licenses for the paid packages that I am able to receive.
On the other hand, I can write my own code, either stand-alone or interop with STK, but I lack knowledge of the math involved.
Additionally, how can I calculate the impact velocity and the maximum extent of the debris field in the event of a successful intercept?