If a particle moves along a straight line with position function s(t) = t^3 - 6t^2 + 9t + 2 (in meters, t in seconds), at what times is the particle momentarily at rest?

Answer and explanation

Correct answer: C. t = 1 s and t = 3 s

  1. Step 1: A particle is momentarily at rest when its instantaneous velocity v(t) = 0.
  2. Step 2: Velocity is the derivative of position: v(t) = s'(t) = 3t^2 - 12t + 9.
  3. Step 3: Set v(t) = 0: 3(t^2 - 4t + 3) = 0 => 3(t - 1)(t - 3) = 0 => t = 1 s and t = 3 s.
Answer
t = 1 s and t = 3 s.
Why not the others: 't = 2 s' is when the acceleration a(t) = s''(t) = 6t - 12 = 0; 't = 0 s and t = 4 s' are arbitrary times with non-zero velocities (9 m/s); 't = 3 s and t = 6 s' factors incorrectly.

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