Home›Calculus 3›Velocity, acceleration and speed in space›Problem 9.235

Velocity, acceleration and speed in space

Problem 9.235 · medium

A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle 0, 0, -32 \right\rangle \), initial velocity \( \displaystyle \left\langle -5, 3, 3 \right\rangle \) and initial position \( \displaystyle \left\langle 0, -3, -1 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).

\[ \left[\begin{matrix}\frac{d}{d t} \left(-5\right)\\\frac{d}{d t} 3\\\frac{d}{d t} \left(3 - 32 t\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\-32\end{matrix}\right] \]
Integrating a and using v(0) gives v(t) = <-5, 3, 3 - 32*t>.✓ Proved
\[ \left[\begin{matrix}\frac{d}{d t} \left(- 5 t\right)\\\frac{d}{d t} \left(3 t - 3\right)\\\frac{d}{d t} \left(- 16 t^{2} + 3 t - 1\right)\end{matrix}\right] = \left[\begin{matrix}-5\\3\\3 - 32 t\end{matrix}\right] \]
Integrate again and use r(0).✓ Proved
\[ \left[\begin{matrix}0\\-3\\-1\end{matrix}\right] \]
r(0) is the given position.✓ Proved

Answer \( \mathbf r(t) = \left\langle - 5 t, 3 t - 3, - 16 t^{2} + 3 t - 1 \right\rangle \)

✓ 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: pass
qwen3.6:27b-mlx: pass — 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.

Every verdict on record (4)

qwen3.6:27b-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: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04
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.