I am trying to compute what vectors lie in a set which is determined by an event. I have computed the set analytically and can evaluate its size. I need to compare the size of the set with some theoretical constant and they must agree. The theoretical constant is about $11.5$. However, I keep computing a much lower value of $4.0$ which. However, adding a minus sign in a place I believe shouldn't be there gives my the right result. I would very much appreciate if people could critique my analysis below and tell me if it is wrong or not. I have given my final solution and the solution that agrees with the theoretical constant.
This questions has ideas similar to that found in: How to find the intersection point of two moving spheres? .
The set up. I have three moving spheres of diameter 1 call them $A,B$ and $C$. Given that $A$ and $B$ collide and $\tau$ seconds later $A$ and $C$ collide. The velocities of $A$ and $C$ call them $v_A$ and $v_C$ are given. Furthermore, when the spheres $A$ and $B$ initially touch the unit vector between their origins is $\nu_{BA}$ pointing from the centre of $B$ to the centre of $A$ and when the spheres $A$ and $C$ initially touch the unit vector between their origins is $\nu_{CA}$ pointing from the centre of $C$ to the centre of $A$, both these vectors are given.
For what velocities $v_B$ do we have that spheres $B$ and $C$ initially touch before $A$ and $B$ touch, this also implies that $B$ and $C$ touch before $A$ and $C$ touch. Furthermore, it does not matter when these two spheres $B$ and $C$ stop overlapping each other.
To summarise. Given $v_A,v_C,\tau,\nu_{BA},\nu_{CA}$ what velocities $v_B$ cause the above event to occur.
My approach:
Suppose particles $A$ and $B$ touch at time $t=0$ then particles $A$ and $C$ touch at time $t= \tau$. Furthermore, suppose that $B$ is at the origin at time zero. Finally, the initial position of particle $C$ is given by $\delta:= \tau(v_A-v_C) + \nu_{BA} - \nu_{CA}$
Know we can define the position vectors of each particle at time $t$.
\begin{align*} X_A(t) &= \nu_{BA} + tv_A,\\ X_B(t) &= tv_B,\\ X_C(t) &= \delta + tv_A. \end{align*}
Then we require to solve $|X_B(t)-X_C(t)|=|t(v_B-v_C) - \delta|=1$. Since we want $B$ and $C$ to initially touch before $A$ and $B$ touch i.e. before $t=0$. Thus by solving $|X_B(t)-X_C(t)|^2=1$ we get two solutions,
\begin{equation} s_\pm = \frac{\langle v_B-v_C, \delta\rangle \pm \sqrt{\langle v_B-v_C, \delta\rangle^2 - |v_B-v_C|^2(|\delta|^2-1)}}{|v_B-v_C|^2} \end{equation}
from here we can conclude that $v_A$ satisfies all the needed criterion if and only if the solutions $s_\pm$ are real and that $s_- \leq 0$ since $s_-$ represents the time the two spheres $B$ and $C$ initially touch. This last condition is equivalent to, \begin{equation} \langle v_B-v_C, \delta\rangle \leq \sqrt{\langle v_B-v_C, \delta\rangle^2 - |v_B-v_C|^2(|\delta|^2-1)}. \end{equation}
However, when comparing the above event to my simulations it seems to condition \begin{equation} -\langle v_B-v_C, \delta\rangle \leq \sqrt{\langle v_B-v_C, \delta\rangle^2 - |v_B-v_C|^2(|\delta|^2-1)}, \end{equation} gives answers that agree with the simulations and not the first case.