A particle starts from rest and moves along a straight path with a time-dependent acceleration given by a(t) = 12t – 6, where t is in seconds and acceleration is in m/s². Calculate the velocity function v(t) by integrating the acceleration and determine the particle’s velocity at t = 6 seconds. Next, obtain the displacement function s(t) by integrating the velocity and find the total displacement covered in the first 5 seconds. Determine the time at which the particle reaches its maximum velocity before deceleration begins and compute the maximum velocity value. Use this to check when the particle comes momentarily to rest again. Explain how the sign change in acceleration indicates a shift from acceleration to deceleration, and why velocity doesn't always imply positive displacement. Discuss the physical relevance of integrating time-varying acceleration in practical motion analysis, such as automotive or robotic applications. Finally, sketch the acceleration, velocity, and displacement curves versus time on a shared axis, clearly labeling the turning point and zero crossings. Comment briefly on how understanding motion through calculus-based kinematics gives designers deeper control over dynamic systems rather than relying solely on constant acceleration formulas.