Velocity, acceleration and speed in space
Problem 9.313 · medium
A particle has position \( \displaystyle \mathbf r(t) = \left\langle t \cos{\left(t \right)}, t \sin{\left(t \right)}, t \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = \frac{\pi}{2} \).
- \[ \left[\begin{matrix}\frac{d}{d t} t \cos{\left(t \right)}\\\frac{d}{d t} t \sin{\left(t \right)}\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}- t \sin{\left(t \right)} + \cos{\left(t \right)}\\t \cos{\left(t \right)} + \sin{\left(t \right)}\\1\end{matrix}\right] \]v = r′.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} \left(- t \sin{\left(t \right)} + \cos{\left(t \right)}\right)\\\frac{d}{d t} \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right)\\\frac{d}{d t} 1\end{matrix}\right] = \left[\begin{matrix}- t \cos{\left(t \right)} - 2 \sin{\left(t \right)}\\- t \sin{\left(t \right)} + 2 \cos{\left(t \right)}\\0\end{matrix}\right] \]a = v′.✓ Proved
- \[ \sqrt{2 + \frac{\pi^{2}}{4}} = \frac{\sqrt{8 + \pi^{2}}}{2} \]Speed = ‖v(pi/2)‖.✓ Proved
Answer \( \mathbf v = \left\langle - \frac{\pi}{2}, 1, 1 \right\rangle,\ \mathbf a = \left\langle -2, - \frac{\pi}{2}, 0 \right\rangle,\ \text{speed} = \frac{\sqrt{8 + \pi^{2}}}{2} \)
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 | difference quotients of the position |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly derives the velocity and acceleration vectors and evaluates them at t = pi/2. The final speed calculation is algebraically correct and matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly derives the velocity and acceleration vectors and evaluates them at t = pi/2. The final speed calculation is algebraically correct and matches the stated answer.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly evaluate the velocity and acceleration vectors at t = pi/2, which is required by the problem statement. While the final speed calculation is correct, the stated answers for v and a are missing from the derivation steps, making the solution incomplete.
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.