Velocity, acceleration and speed in space
Problem 9.238 · medium
A particle has position \( \displaystyle \mathbf r(t) = \left\langle 3 t, 4 t^{2}, 2 t^{3} \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = 2 \).
- \[ \left[\begin{matrix}\frac{d}{d t} 3 t\\\frac{d}{d t} 4 t^{2}\\\frac{d}{d t} 2 t^{3}\end{matrix}\right] = \left[\begin{matrix}3\\8 t\\6 t^{2}\end{matrix}\right] \]v = r′.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} 3\\\frac{d}{d t} 8 t\\\frac{d}{d t} 6 t^{2}\end{matrix}\right] = \left[\begin{matrix}0\\8\\12 t\end{matrix}\right] \]a = v′.✓ Proved
- \[ 29 \]Speed = ‖v(2)‖.✓ Proved
Answer \( \mathbf v = \left\langle 3, 16, 24 \right\rangle,\ \mathbf a = \left\langle 0, 8, 24 \right\rangle,\ \text{speed} = 29 \)
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: fail (error) — The solution fails to explicitly evaluate the velocity and acceleration vectors at t=2 before stating the final answers. While the final numerical answers are correct, the steps shown only provide the general symbolic expressions for v(t) and a(t), making the derivation of the specific values at t=2 logically incomplete.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly evaluate the velocity and acceleration vectors at t=2 before stating the final answers. While the final numerical answers are correct, the steps shown only provide the general symbolic expressions for v(t) and a(t), making the derivation of the specific values at t=2 logically incomplete.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly evaluate the velocity and acceleration vectors at t=2 before stating the final answer. While the final numerical values are correct, the intermediate steps (lines 1 and 2) only provide the general symbolic expressions for v(t) and a(t). A student might incorrectly assume that the general expressions are the answers or miss the crucial step of substitution. The solution should explicitly show v(2) and a(2).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.