Velocity, acceleration and speed in space
Problem 9.237 · medium
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle 0, 0, -32 \right\rangle \), initial velocity \( \displaystyle \left\langle 5, -1, -3 \right\rangle \) and initial position \( \displaystyle \left\langle 1, -3, 0 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
- \[ \left[\begin{matrix}\frac{d}{d t} 5\\\frac{d}{d t} \left(-1\right)\\\frac{d}{d t} \left(- 32 t - 3\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\-32\end{matrix}\right] \]Integrating a and using v(0) gives v(t) = <5, -1, -32*t - 3>.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} \left(5 t + 1\right)\\\frac{d}{d t} \left(- t - 3\right)\\\frac{d}{d t} \left(- 16 t^{2} - 3 t\right)\end{matrix}\right] = \left[\begin{matrix}5\\-1\\- 32 t - 3\end{matrix}\right] \]Integrate again and use r(0).✓ Proved
- \[ \left[\begin{matrix}1\\-3\\0\end{matrix}\right] \]r(0) is the given position.✓ Proved
Answer \( \mathbf r(t) = \left\langle 5 t + 1, - t - 3, t \left(- 16 t - 3\right) \right\rangle \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the motion integrated numerically from the initial conditions, compared at t = 1.5 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly integrates acceleration to find velocity and then velocity to find position, applying the initial conditions accurately. The final vector matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly integrates acceleration to find velocity and then velocity to find position, applying the initial conditions accurately. The final vector matches the stated answer.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly integrates the acceleration vector to find velocity, applies the initial velocity condition, and then integrates velocity to find position, applying the initial position condition. The final result matches the stated answer.gpt-oss:20b: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The third component of the position vector is incorrect. Integrating the velocity component \(-32t-3\) gives \(-16t^2-3t+C\), and using \(r(0)=\langle1,-3
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/motion_in_space, checked 2026-10-04 with SymPy 1.14.0.