Particle motion: position, velocity, acceleration
Problem 2.1721 · hard
A particle moves along a line with position \( \displaystyle s(t) = t^{3} - 6 t^{2} + 9 t - 3 \) (meters, \( \displaystyle t \) in seconds). Find its velocity and acceleration, the times it is at rest, and the total distance it travels for \( \displaystyle 0 \le t \le 5 \).
- \[ \frac{d}{d t} \left(t^{3} - 6 t^{2} + 9 t - 3\right) = 3 t^{2} - 12 t + 9 \]Velocity is the derivative of position.✓ Proved
- \[ \frac{d}{d t} \left(3 t^{2} - 12 t + 9\right) = 6 t - 12 \]Acceleration is the derivative of velocity.✓ Proved
- \[ 3 t^{2} - 12 t + 9 = \left(t - 1\right) \left(3 t - 9\right) \]v = 0 at t = 1 and t = 3: the particle is at rest.✓ Proved
- v changes sign at each of those times, so the particle turns around there. Distance adds the length of each leg: [0, 1], [1, 3], [3, 5].Reviewed
- \[ 4 + 4 + 20 = 28 \]|s(1) − s(0)| + |s(3) − s(1)| + |s(5) − s(3)|, with s = -3, 1, -3, 17.✓ Proved
- For comparison, the displacement is s(5) − s(0) = 20 m.Reviewed
Answer \( v(t) = 3 t^{2} - 12 t + 9,\ a(t) = 6 t - 12,\ \text{at rest at } t = 1, 3,\ \text{distance} = 28\text{ m} \)
✓ Nihil obstat Lines: 4 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | integral of |v(t)| by numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies velocity, acceleration, and rest times. The calculation of total distance by summing absolute displacements over intervals where velocity does not change sign is correct.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/particle_motion, checked 2026-10-06 with SymPy 1.14.0.