A particle moving along the x axis has a position given by x = 54t − 2.0t³ m. At the time t = 3.0 s, the speed of the particle is zero. Which statement is correct? A) The particle remains at rest after t = 3.0 s. B) The particle no longer accelerates after t = 3.0 s. C) The particle can be found at positions x

Question

A particle moving along the x axis has a position given by x = 54t − 2.0t³ m. At the time t = 3.0 s, the speed of the particle is zero. Which statement is correct? A) The particle remains at rest after t = 3.0 s. B) The particle no longer accelerates after t = 3.0 s. C) The particle can be found at positions x < 0 m only when t < 0 s. D) All of the above are correct. E) None of the above is correct.

Quizplus · Accepted Answer

The given position function is $x = 54t - 2.0t^3$ meters. The speed of the particle is the magnitude of the velocity, which is the first derivative of the position with respect to time, $v = dx/dt = 54 - 6t^2$. Setting this equal to zero to find when the speed is zero gives $54 - 6t^2 = 0$, which has a real solution, indicating the particle does stop at some point (t = 3s as given), but it does not imply the particle remains at rest after t = 3.0 s, nor does it imply the particle no longer accelerates after t = 3.0 s (since acceleration, the second derivative of position, is constant and non-zero, $a = d^2x/dt^2 = -12t$). The particle can be found at positions x < 0 for values of t that make $54t - 2.0t^3 < 0$, which is not limited to t < 0 s. Therefore, none of the given statements are correct.

A Particle Moving Along the X Axis Has a Position