Final answer:
Acceleration indicates the direction of change in velocity relative to a coordinate system; a particle can move and even accelerate in any direction regardless of whether the acceleration is positive or negative. The correct relationship is that the direction of motion depends on the initial velocity and acceleration together, making option 'c' the right answer.
Step-by-step explanation:
The question is about the relationship between the direction of a particle's motion and its acceleration. Acceleration is defined as the rate of change in velocity, which includes both magnitude and direction. It is critical to understand that positive or negative acceleration does not inherently determine the direction of motion; it merely indicates the direction of velocity change relative to a chosen coordinate system.
If a particle is moving to the right (positive direction) and speeding up, it has a positive acceleration. If the same particle is moving to the right but slowing down, it has a negative acceleration because its acceleration is toward the left or opposite the motion. In contrast, a particle moving to the left (negative direction) and slowing down is experiencing positive acceleration because its velocity is becoming less negative over time (in the positive direction). Finally, a particle moving to the left and speeding up has a negative acceleration, as its velocity is becoming more negative.
Therefore, the correct answer regarding the relationship between motion and acceleration is: c) Direction depends on initial velocity. This is because acceleration impacts the change in velocity, not the absolute direction of travel. Thus, positive acceleration does not always imply rightward motion, nor does negative acceleration always imply leftward motion. The initial velocity's direction, along with acceleration, determines the overall motion of the particle.