Particle motion: position, velocity, acceleration

Problem 2.1872 · hard

1. \[ \frac{d}{d t} \left(t^{3} - 12 t^{2} + 45 t + 2\right) = 3 t^{2} - 24 t + 45 \]
   Velocity is the derivative of position.✓ Proved
2. \[ \frac{d}{d t} \left(3 t^{2} - 24 t + 45\right) = 6 t - 24 \]
   Acceleration is the derivative of velocity.✓ Proved
3. \[ 3 t^{2} - 24 t + 45 = \left(t - 3\right) \left(3 t - 15\right) \]
   v = 0 at t = 3 and t = 5: 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, 3], [3, 5], [5, 6].
   Reviewed
5. \[ 4 + 4 + 54 = 62 \]
   |s(3) − s(0)| + |s(5) − s(3)| + |s(6) − s(5)|, with s = 2, 56, 52, 56.✓ Proved
6. For comparison, the displacement is s(6) − s(0) = 54 m.
   Reviewed