Particle motion: position, velocity, acceleration

Problem 2.1723 · hard

1. \[ \frac{d}{d t} \left(t^{3} - 9 t^{2} + 15 t + 1\right) = 3 t^{2} - 18 t + 15 \]
   Velocity is the derivative of position.✓ Proved
2. \[ \frac{d}{d t} \left(3 t^{2} - 18 t + 15\right) = 6 t - 18 \]
   Acceleration is the derivative of velocity.✓ Proved
3. \[ 3 t^{2} - 18 t + 15 = \left(t - 1\right) \left(3 t - 15\right) \]
   v = 0 at t = 1 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, 1], [1, 5], [5, 7].
   Reviewed
5. \[ 7 + 32 + 32 = 71 \]
   |s(1) − s(0)| + |s(5) − s(1)| + |s(7) − s(5)|, with s = 1, 8, -24, 8.✓ Proved
6. For comparison, the displacement is s(7) − s(0) = 7 m.
   Reviewed