Consider two particles, each of mass m , positioned at a separation...

The magnetic force on a moving charge is given by $\vec{F} = q(\vec{v} \times \vec{B})$. For the particle with charge $q$ moving with velocity $v\hat{i}$ in a magnetic field $-B\hat{k}$, the force is $qvB\hat{j}$ (upwards). For the particle with charge $-q$ moving with velocity $-v\hat{i}$, the force is also $qvB\hat{j}$ (upwards). Thus, both particles move in circular paths curving upwards.

The radius of the circular path is $R = \dfrac{mv}{qB}$.

To just avoid collision, the two circular paths must be tangent to each other. Since the initial separation is $d$, the sum of their radii must be equal to $d$.

$2R_m = d \Rightarrow R_m = \dfrac{d}{2}$

The maximum speed to avoid collision is $v_m = \dfrac{qBR_m}{m} = \dfrac{qBd}{2m}$.

When the particles are projected with speed $v = 2v_m$, the new radius of their paths is $R = \dfrac{m(2v_m)}{qB} = 2R_m = d$.

Let the collision occur after the velocity vectors have rotated by an angle $\theta$. The horizontal distance covered by each particle is $\dfrac{d}{2}$.

From the geometry of the circular path, the horizontal displacement is $R \sin\theta$.

$R \sin\theta = \dfrac{d}{2}$

Substituting $R = d$, we get $\sin\theta = \dfrac{1}{2} \Rightarrow \theta = 30^{\circ}$.

The velocities of the particles just before the collision are:

$\vec{v}_1 = v \cos(30^{\circ})\hat{i} + v \sin(30^{\circ})\hat{j}$

$\vec{v}_2 = -v \cos(30^{\circ})\hat{i} + v \sin(30^{\circ})\hat{j}$

Since the collision is completely inelastic, the particles stick together. By conservation of linear momentum:

$m\vec{v}_1 + m\vec{v}_2 = (2m)\vec{v}_f$

$\vec{v}_f = \dfrac{\vec{v}_1 + \vec{v}_2}{2} = v \sin(30^{\circ})\hat{j} = \dfrac{v}{2}\hat{j}$

Given $v = 2v_m$, the final velocity is $\vec{v}_f = v_m\hat{j}$.

The combined mass has a net charge of $q + (-q) = 0$. Therefore, it experiences no magnetic force and moves with a constant speed $v_m$ along a straight line drawn upward.

Answer: It moves with a constant speed $v_m$ along a straight line drawn upward in the plane of the figure through the point of collision.