Final answer:
To move with a constant velocity, a particle must be projected with a minimum initial speed so that the force exerted on it by velocity and a constant horizontal vector counterbalances the force of gravity. The minimum speed is given by v = mg/A, where g is the acceleration due to gravity.
Step-by-step explanation:
In this problem, we can utilize the force equation F = ma, where m is the mass of the particle and a is its acceleration. We know that the total force acting on the particle is the sum of the force of gravity and the force F, such that F + mg = ma. This must equal zero because the particle moves at a constant velocity (hence, acceleration a = 0). Accordingly, from the equation F = mg, we deduce that the force F must cancel out the force of gravity for the particle to move with a constant velocity.
The force F is given as F = v × A. As a result, for the particle to move at a constant velocity, its minimum initial speed must be such that F cancels out the force of gravity. By doing so, we can solve for v: v = mg/A, where g is the acceleration due to gravity.