Velocity, acceleration and speed in space
Problem 9.233 · medium
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle \cos{\left(t \right)}, - \sin{\left(t \right)}, 1 \right\rangle \), initial velocity \( \displaystyle \left\langle 3, 1, -5 \right\rangle \) and initial position \( \displaystyle \left\langle -2, -1, 3 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
- \[ \left[\begin{matrix}\frac{d}{d t} \left(\sin{\left(t \right)} + 3\right)\\\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \left(t - 5\right)\end{matrix}\right] = \left[\begin{matrix}\cos{\left(t \right)}\\- \sin{\left(t \right)}\\1\end{matrix}\right] \]Integrating a and using v(0) gives v(t) = <sin(t) + 3, cos(t), t - 5>.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} \left(3 t - \cos{\left(t \right)} - 1\right)\\\frac{d}{d t} \left(\sin{\left(t \right)} - 1\right)\\\frac{d}{d t} \left(\frac{t^{2}}{2} - 5 t + 3\right)\end{matrix}\right] = \left[\begin{matrix}\sin{\left(t \right)} + 3\\\cos{\left(t \right)}\\t - 5\end{matrix}\right] \]Integrate again and use r(0).✓ Proved
- \[ \left[\begin{matrix}-2\\-1\\3\end{matrix}\right] \]r(0) is the given position.✓ Proved
Answer \( \mathbf r(t) = \left\langle 3 t - \cos{\left(t \right)} - 1, \sin{\left(t \right)} - 1, \frac{t^{2}}{2} - 5 t + 3 \right\rangle \)
✓ Nihil obstat 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. 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 |
| 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, applies the initial velocity condition, integrates velocity to find position, and applies the initial position condition. The final result 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, applies the initial velocity condition, integrates velocity to find position, and applies the initial position condition. The final result matches the stated answer.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly integrates acceleration to find velocity and then position, applying the initial conditions accurately. The final vector function matches the stated answer.gpt-oss:20b: pass 2026-10-04
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.