The velocity of a particle moving along the x axis is given for t 0 by v_x = (32.0t − 2.00t³) m/s, where t is in s. What is the acceleration of the particle when (after t = 0) it achieves its maximum displacement in the positive x direction? A) −64.0 m/s² B) zero C) 128 m/s² D) 32.0 m/s² E) −32.0 m/s²

Question

The velocity of a particle moving along the x axis is given for t > 0 by v_x = (32.0t − 2.00t³) m/s, where t is in s. What is the acceleration of the particle when (after t = 0) it achieves its maximum displacement in the positive x direction? A) −64.0 m/s² B) zero C) 128 m/s² D) 32.0 m/s² E) −32.0 m/s²

Quizplus · Accepted Answer

The maximum displacement in the positive x direction occurs when the velocity is zero, as this is the turning point where the particle stops moving forward and starts moving back. To find when this occurs, set the velocity equation to zero and solve for t: 0 = 32.0t - 2.00t^3. This gives a non-zero t value at which the velocity is zero. The acceleration is the derivative of the velocity with respect to time, a = dv/dt = 32.0 - 6.00t^2. Substituting the t value found from setting the velocity to zero into the acceleration equation will give the acceleration at the point of maximum displacement. Since the acceleration is negative (as indicated by the correct choice), it shows that the particle is decelerating at the maximum displacement point in the positive x direction.

The Velocity of a Particle Moving Along the X Axis