Particle motion: position, velocity, acceleration

Problem 2.1871 · hard

1. \[ \frac{d}{d t} \left(t^{3} - 12 t^{2} + 36 t + 6\right) = 3 t^{2} - 24 t + 36 \]
   Velocity is the derivative of position.✓ Proved
2. \[ \frac{d}{d t} \left(3 t^{2} - 24 t + 36\right) = 6 t - 24 \]
   Acceleration is the derivative of velocity.✓ Proved
3. \[ 3 t^{2} - 24 t + 36 = \left(t - 2\right) \left(3 t - 18\right) \]
   v = 0 at t = 2 and t = 6: the particle is at rest.✓ Proved
4. v changes sign at each of those times, so the particle turns around there. Distance adds the length of each leg: [0, 2], [2, 6], [6, 8].
   Reviewed
5. \[ 32 + 32 + 32 = 96 \]
   |s(2) − s(0)| + |s(6) − s(2)| + |s(8) − s(6)|, with s = 6, 38, 6, 38.✓ Proved
6. For comparison, the displacement is s(8) − s(0) = 32 m.
   Reviewed