In addition to the force of gravity, a force F = v × A acts on a...

For the particle to move undeflected at a constant velocity, the net force acting on it must be zero.

The forces acting on the particle are the gravitational force $m\vec{g}$ and the given force $\vec{F} = \vec{v} \times \vec{A}$.

Equating the net force to zero gives:

$m\vec{g} + \vec{v} \times \vec{A} = 0$

$\vec{v} \times \vec{A} = -m\vec{g}$

Taking the magnitude on both sides yields:

$v A \sin \theta = mg$

where $\theta$ is the angle between the velocity vector $\vec{v}$ and the constant horizontal vector $\vec{A}$.

To minimize the speed $v$, $\sin \theta$ must be maximized. The maximum value of $\sin \theta$ is $1$, which occurs when $\theta = 90^{{}^{{}^{\circ}}}$.

Substituting $\sin \theta = 1$ gives:

v_\min A = mg

v_\min = \dfrac{mg}{A}