Velocity, acceleration and speed in space
Problem 9.471 · 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 \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\frac{d}{d t} 3 t\\\frac{d}{d t} 4 t^{2}\\\frac{d}{d t} 2 t^{3}\end{matrix}\right] = \left[\begin{matrix}3\\8 t\\6 t^{2}\end{matrix}\right] \]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
- \[ \sqrt{109} \]Speed = ‖v(1)‖.✓ Proved
Answer \( \mathbf v = \left\langle 3, 8, 6 \right\rangle,\ \mathbf a = \left\langle 0, 8, 12 \right\rangle,\ \text{speed} = \sqrt{109} \)
✓ 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 | difference quotients of the position |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the derivatives for velocity and acceleration, and correctly evaluates the speed at t=1. The steps are logically sound and algebraically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly computes the derivatives for velocity and acceleration, and correctly evaluates the speed at t=1. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly computes the velocity and acceleration vectors by differentiation and evaluates them at t=1 to find the speed. The steps are logically sound and the final answer matches the stated result.
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 2026-10-10 with SymPy 1.14.0.