Velocity, acceleration and speed in space
Problem 9.555 · medium
A particle has position \( \displaystyle \mathbf r(t) = \left\langle e^{t}, e^{- t}, t \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\frac{d}{d t} e^{t}\\\frac{d}{d t} e^{- t}\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}e^{t}\\- e^{- t}\\1\end{matrix}\right] \]v = r′.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d t} e^{t}\\\frac{d}{d t} \left(- e^{- t}\right)\\\frac{d}{d t} 1\end{matrix}\right] = \left[\begin{matrix}e^{t}\\e^{- t}\\0\end{matrix}\right] \]a = v′.✓ Proved
- \[ \sqrt{e^{-2} + 1 + e^{2}} = \frac{\sqrt{1 + e^{2} + e^{4}}}{e} \]Speed = ‖v(1)‖.✓ Proved
Answer \( \mathbf v = \left\langle e, - \frac{1}{e}, 1 \right\rangle,\ \mathbf a = \left\langle e, e^{-1}, 0 \right\rangle,\ \text{speed} = \frac{\sqrt{1 + e^{2} + e^{4}}}{e} \)
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 computes the velocity and acceleration vectors by differentiating the position vector. The speed calculation at t=1 is algebraically correct, as the expression simplifies to the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly computes the velocity and acceleration vectors by differentiating the position vector. The speed calculation at t=1 is algebraically correct, as the expression simplifies to the stated answer.gpt-oss:20b: fail (misleading) 2026-10-12 — The speed calculation is incorrect: it uses √(e^{-2}+1+e^{2})e^{-1}, which does not equal the norm of the velocity vector at t=1. The correct speed is √(e^{2}+e^{-2}+1).qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly computes the velocity and acceleration vectors by differentiating the position vector. The speed is correctly calculated as the magnitude of the velocity vector at t=1, and the algebraic simplification in the final line is correct.
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.