Velocity, acceleration and speed in space
Problem 9.319 · medium
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle \cos{\left(t \right)}, - \sin{\left(t \right)}, 1 \right\rangle \), initial velocity \( \displaystyle \left\langle -1, 3, 4 \right\rangle \) and initial position \( \displaystyle \left\langle -3, -2, 1 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
- \[ \left[\begin{matrix}\frac{d}{d t} \left(\sin{\left(t \right)} - 1\right)\\\frac{d}{d t} \left(\cos{\left(t \right)} + 2\right)\\\frac{d}{d t} \left(t + 4\right)\end{matrix}\right] = \left[\begin{matrix}\cos{\left(t \right)}\\- \sin{\left(t \right)}\\1\end{matrix}\right] \]Integrating a and using v(0) gives v(t) = <sin(t) - 1, cos(t) + 2, t + 4>.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} \left(- t - \cos{\left(t \right)} - 2\right)\\\frac{d}{d t} \left(2 t + \sin{\left(t \right)} - 2\right)\\\frac{d}{d t} \left(\frac{t^{2}}{2} + 4 t + 1\right)\end{matrix}\right] = \left[\begin{matrix}\sin{\left(t \right)} - 1\\\cos{\left(t \right)} + 2\\t + 4\end{matrix}\right] \]Integrate again and use r(0).✓ Proved
- \[ \left[\begin{matrix}-3\\-2\\1\end{matrix}\right] \]r(0) is the given position.✓ Proved
Answer \( \mathbf r(t) = \left\langle - t - \cos{\left(t \right)} - 2, 2 t + \sin{\left(t \right)} - 2, \frac{t^{2}}{2} + 4 t + 1 \right\rangle \)
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: fail (error) — The solution contains a sign error in the y-component of the velocity. The integral of -sin(t) is cos(t), so v_y(t) should be cos(t) + 3, not cos(t) + 2. This leads to an incorrect final position vector.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution contains a sign error in the y-component of the velocity. The integral of -sin(t) is cos(t), so v_y(t) should be cos(t) + 3, not cos(t) + 2. This leads to an incorrect final position vector.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.