Consider two particles, A and B, moving in $\mathbb{R}^3$. Their position vectors, $\mathbf{r}_A(t)$ and $\mathbf{r}_B(t)$, respectively, are governed by the following second-order ordinary differential equations (ODEs):
$\frac{d^2 \mathbf{r}_A}{dt^2} = \mathbf{a}_A$ $\frac{d^2 \mathbf{r}_B}{dt^2} = \mathbf{a}_B$
where $\mathbf{a}A$ and $\mathbf{a}_B$ are constant acceleration vectors in $\mathbb{R}^3$. Let $\mathbf{r}_A(t) = (x{1A}(t), x_{2A}(t), x_{3A}(t))$ and $\mathbf{r}B(t) = (x{1B}(t), x_{2B}(t), x_{3B}(t))$.
Let $\mathbf{r}{A0} = \mathbf{r}_A(0)$, $\mathbf{v}{A0} = \frac{d\mathbf{r}A}{dt}(0)$ denote the initial position and velocity vectors of particle A, respectively. Similarly, let $\mathbf{r}{B0} = \mathbf{r}B(0)$, $\mathbf{v}{B0} = \frac{d\mathbf{r}_B}{dt}(0)$ denote the initial position and velocity vectors of particle B, respectively.
Furthermore, let $t^ \in \mathbb{R}$, $t^ > 0$, be the time at which the two particles occupy the same position; i.e., $\mathbf{r}A(t^) = \mathbf{r}_B(t^)$. We constrain the initial velocity of particle B such that $\lVert \mathbf{v}{B0} \rVert = s_b$, where $s_b$ is a constant scalar.
Determine the symbolic expressions for the components of the initial velocity vector $\mathbf{v}{B0}$, denoted by $v{b1}$, $v_{b2}$, $v_{b3}$, and the time of coincidence $t^*$, given the initial conditions, the constant accelerations, and the velocity magnitude constraint for particle B. Express the general solution in terms of known quantities.
Given the constant acceleration, and defining $\mathbf{v}_A(t) = \frac{d\mathbf{r}_A}{dt}$, $\mathbf{v}_B(t) = \frac{d\mathbf{r}_B}{dt}$, the position vectors of the particles are:
$$ \mathbf{r}A(t) = \mathbf{r}{A0} + \mathbf{v}{A0} t + \frac{1}{2} \mathbf{a}_A t^2 $$ $$ \mathbf{r}_B(t) = \mathbf{r}{B0} + \mathbf{v}_{B0} t + \frac{1}{2} \mathbf{a}_B t^2 $$
Where: $\mathbf{v}A(t) = \mathbf{v}{A0} + \mathbf{a}A t$ and $\mathbf{v}_B(t) = \mathbf{v}{B0} + \mathbf{a}_B t$
The particles coincide when $\mathbf{r}A(t^) = \mathbf{r}_B(t^)$, which expands to: $$ \mathbf{r}{A0} + \mathbf{v}{A0} t^ + \frac{1}{2} \mathbf{a}_A (t^)^2 = \mathbf{r}{B0} + \mathbf{v}_{B0} t^ + \frac{1}{2} \mathbf{a}_B (t^)^2 $$
The magnitude of the initial velocity vector $\mathbf{v}_{B0}$ is constrained by $s_b$:
$$ \lVert \mathbf{v}_{B0} \rVert = s_b $$
$$ \frac{dx_{1A}}{dt}(0) t^ + \frac{1}{2} \frac{d^2 x_{1A}}{dt^2}(0) (t^)^2 = x_{1B}(0) + \frac{dx_{1B}}{dt}(0) t^ + \frac{1}{2} \frac{d^2 x_{1B}}{dt^2}(0) (t^)^2 $$ $$ x_{2A}(0) + \frac{dx_{2A}}{dt}(0) t^ + \frac{1}{2} \frac{d^2 x_{2A}}{dt^2}(0) (t^)^2 = x_{2B}(0) + \frac{dx_{2B}}{dt}(0) t^ + \frac{1}{2} \frac{d^2 x_{2B}}{dt^2}(0) (t^)^2 $$ $$ x_{3A}(0) + \frac{dx_{3A}}{dt}(0) t^ + \frac{1}{2} \frac{d^2 x_{3A}}{dt^2}(0) (t^)^2 = x_{3B}(0) + \frac{dx_{3B}}{dt}(0) t^ + \frac{1}{2} \frac{d^2 x_{3B}}{dt^2}(0) (t^)^2 $$
$$ v_{b1}^2 + v_{b2}^2 + v_{b3}^2 = s_b^2 $$ Where: $v_{b1}$, $v_{b2}$, and $v_{b3}$ are the components of $\mathbf{v}{B0}$. The components of the acceleration vectors are $\left( \frac{d^2 x{1A}}{dt^2}, \frac{d^2 x_{2A}}{dt^2}, \frac{d^2 x_{3A}}{dt^2} \right) = \mathbf{a}A$ and $\left( \frac{d^2 x{1B}}{dt^2}, \frac{d^2 x_{2B}}{dt^2}, \frac{d^2 x_{3B}}{dt^2} \right) = \mathbf{a}_B$
If closed-form solutions are not feasible, discuss methods to numerically approximate the solution to this problem.