Velocity, acceleration and speed in space
Problem 9.551 · medium
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle 6 t, 2, 0 \right\rangle \), initial velocity \( \displaystyle \left\langle -4, -2, 2 \right\rangle \) and initial position \( \displaystyle \left\langle -1, 2, 2 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
- \[ \left[\begin{matrix}\frac{d}{d t} \left(3 t^{2} - 4\right)\\\frac{d}{d t} \left(2 t - 2\right)\\\frac{d}{d t} 2\end{matrix}\right] = \left[\begin{matrix}6 t\\2\\0\end{matrix}\right] \]Integrating a and using v(0) gives v(t) = <3*t**2 - 4, 2*t - 2, 2>.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} \left(t^{3} - 4 t - 1\right)\\\frac{d}{d t} \left(t^{2} - 2 t + 2\right)\\\frac{d}{d t} \left(2 t + 2\right)\end{matrix}\right] = \left[\begin{matrix}3 t^{2} - 4\\2 t - 2\\2\end{matrix}\right] \]Integrate again and use r(0).✓ Proved
- \[ \left[\begin{matrix}-1\\2\\2\end{matrix}\right] \]r(0) is the given position.✓ Proved
Answer \( \mathbf r(t) = \left\langle t^{3} - 4 t - 1, t^{2} - 2 t + 2, 2 t + 2 \right\rangle \)
✓ 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 and then position, applying the initial conditions accurately at each step.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly integrates acceleration to find velocity and then position, applying the initial conditions accurately at each step.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.