Final answer:
A particle with positive and increasing acceleration on the x-axis will have a velocity that not only increases but does so at a quicker rate as time progresses, resulting in the particle speeding up to the right more rapidly.
Step-by-step explanation:
When a particle's acceleration is positive and increasing while it moves along the x-axis, this indicates that the particle's velocity is also increasing in the positive or rightward direction. An increasing positive acceleration means the rate at which the particle is speeding up is itself becoming faster over time. As a result, not only is the particle moving faster and faster (speeding up), but the pace at which it speeds up is accelerating.
If we consider an object like a train starting from rest, with a positive and increasing acceleration, initially, it picks up speed gradually. Over time, as the acceleration continues to increase, the gains in velocity become more pronounced. This concept aligns with the data from a velocity vs. time graph where a positively sloped line indicates positive velocity, and if that slope increases, it reflects increasing acceleration.
In summary, a positive, increasing acceleration affects the particle by causing it to move rightward on the x-axis with a velocity that grows more rapidly as time progresses. This continues until either the acceleration changes or an external force is applied that alters this behavior.