Particle motion: position, velocity, acceleration

Problem 2.1611 · hard

1. \[ \frac{d}{d t} \left(2 t^{3} - 24 t^{2} + 72 t + 9\right) = 6 t^{2} - 48 t + 72 \]
   Velocity is the derivative of position.✓ Proved
2. \[ \frac{d}{d t} \left(6 t^{2} - 48 t + 72\right) = 12 t - 48 \]
   Acceleration is the derivative of velocity.✓ Proved
3. \[ 6 t^{2} - 48 t + 72 = \left(t - 2\right) \left(6 t - 36\right) \]
   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, 7].
   Reviewed
5. \[ 14 + 64 + 64 = 142 \]
   |s(2) − s(0)| + |s(6) − s(2)| + |s(7) − s(6)|, with s = 9, 73, 9, 23.✓ Proved
6. For comparison, the displacement is s(7) − s(0) = 14 m.
   Reviewed