Final answer:
The force needed to maintain the conveyor belt's constant velocity due to the vertical impact of the grain falling at a rate of 19 kg/s at a constant horizontal speed of 4 m/s is 76 N.
Step-by-step explanation:
To answer the question, we must first understand that the force required to keep the conveyor belt moving at a constant velocity is related to the change in momentum of the grain as it falls onto the belt. Here, we are not asked to account for friction between the conveyor belt and the grain, or any resistance in the conveyor's mechanism since the question focuses on the force needed to maintain its velocity due to the impact of the grain.
Since the conveyor belt maintains a constant speed, we are dealing with an inelastic collision where the grain sticks to the conveyor and moves with it. The rate of change of momentum of the grain, which is in the same direction as the conveyor belt speed, will provide us with the force required by the motor to keep the conveyor moving at constant velocity.
The change in momentum per second (which is the force) due to the grain is equal to the mass flow rate of the grain times the velocity of the conveyor belt: F = mv, where 'm' is the mass flow rate in kg/s and 'v' is the velocity of the belt in m/s. For a mass flow rate of 19 kg/s and a velocity of 4 m/s, the force F = 19 kg/s × 4 m/s = 76 N.