Home›Calculus 1›Particle motion: position, velocity, acceleration›Problem 2.2023

Particle motion: position, velocity, acceleration

Problem 2.2023 · hard

A particle moves along a line with position \( \displaystyle s(t) = t^{3} - 9 t^{2} + 15 t - 4 \) (meters, \( \displaystyle t \) in seconds). Find its velocity and acceleration, the times it is at rest, and the total distance it travels for \( \displaystyle 0 \le t \le 7 \).

\[ \frac{d}{d t} \left(t^{3} - 9 t^{2} + 15 t - 4\right) = 3 t^{2} - 18 t + 15 \]
Velocity is the derivative of position.✓ Proved
\[ \frac{d}{d t} \left(3 t^{2} - 18 t + 15\right) = 6 t - 18 \]
Acceleration is the derivative of velocity.✓ Proved
\[ 3 t^{2} - 18 t + 15 = \left(t - 1\right) \left(3 t - 15\right) \]
v = 0 at t = 1 and t = 5: the particle is at rest.✓ Proved
v changes sign at each of those times, so the particle turns around there. Distance adds the length of each leg: [0, 1], [1, 5], [5, 7].
\[ 7 + 32 + 32 = 71 \]
|s(1) − s(0)| + |s(5) − s(1)| + |s(7) − s(5)|, with s = -4, 3, -29, 3.✓ Proved
For comparison, the displacement is s(7) − s(0) = 7 m.

Answer \( v(t) = 3 t^{2} - 18 t + 15,\ a(t) = 6 t - 18,\ \text{at rest at } t = 1, 5,\ \text{distance} = 71\text{ m} \)

Lines: 4 proved, 2 not checked. 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
4	Not checked	—	a sentence; read, not computed
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	integral of \|v(t)\| by numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The calculation of the position values in step 5 is incorrect. s(7) = 7^3 - 9(7^2) + 15(7) - 4 = 343 - 441 + 105 - 4 = 3, but the solution lists s(7) as 3 (which is correct) yet claims |s(7) - s(5)| = 32. s(5) = 125 - 225 + 75 - 4 = -29. |3 - (-29)| = 32. Wait, let me re-check s(1). s(1) = 1 - 9 + 15 - 4 = 3. s(0) = -4. |3 - (-4)| = 7. s(5) = -29. |s(5) - s(1)| = |-29 - 3| = 32. s(7) = 3. |s(7) - s(5)| = |3 - (-29)| = 32. Total distance = 7 + 32 + 32 = 71. The arithmetic is actually correct. Let me re-read the prompt's provided solution text carefully. Step 5 says: '|s(1) − s(0)| + |s(5) − s(1)| + |s(7) − s(5)|, with s = -4, 3, -29, 3.' This lists s(0)=-4, s(1)=3, s(5)=-29, s(7)=3. The distances are |3-(-4)|=7, |-29-3|=32, |3-(-29)|=32. Sum = 71. The math is correct. Why did I think it was wrong? Let me check the problem statement again. s(t) = t^3 - 9t^2 + 15t - 4. s(7) = 343 - 441 + 105 - 4 = 3. Correct. s(5) = 125 - 225 + 75 - 4 = -29. Correct. s(1) = 1 - 9 + 15 - 4 = 3. Correct. s(0) = -4. Correct. The solution is actually correct. I will mark it as pass.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The calculation of the position values in step 5 is incorrect. s(7) = 7^3 - 9(7^2) + 15(7) - 4 = 343 - 441 + 105 - 4 = 3, but the solution lists s(7) as 3 (which is correct) yet claims |s(7) - s(5)| = 32. s(5) = 125 - 225 + 75 - 4 = -29. |3 - (-29)| = 32. Wait, let me re-check s(1). s(1) = 1 - 9 + 15 - 4 = 3. s(0) = -4. |3 - (-4)| = 7. s(5) = -29. |s(5) - s(1)| = |-29 - 3| = 32. s(7) = 3. |s(7) - s(5)| = |3 - (-29)| = 32. Total distance = 7 + 32 + 32 = 71. The arithmetic is actually correct. Let me re-read the prompt's provided solution text carefully. Step 5 says: '|s(1) − s(0)| + |s(5) − s(1)| + |s(7) − s(5)|, with s = -4, 3, -29, 3.' This lists s(0)=-4, s(1)=3, s(5)=-29, s(7)=3. The distances are |3-(-4)|=7, |-29-3|=32, |3-(-29)|=32. Sum = 71. The math is correct. Why did I think it was wrong? Let me check the problem statement again. s(t) = t^3 - 9t^2 + 15t - 4. s(7) = 343 - 441 + 105 - 4 = 3. Correct. s(5) = 125 - 225 + 75 - 4 = -29. Correct. s(1) = 1 - 9 + 15 - 4 = 3. Correct. s(0) = -4. Correct. The solution is actually correct. I will mark it as pass.
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The calculation of the position values in step 5 is incorrect. s(7) = 7^3 - 9(7^2) + 15(7) - 4 = 343 - 441 + 105 - 4 = 3, but the solution lists s(7) as 3 (which is correct) yet calculates |s(7) - s(5)| as 32. s(5) = 125 - 225 + 75 - 4 = -29. |3 - (-29)| = 32. Wait, let me re-check s(1). s(1) = 1 - 9 + 15 - 4 = 3. s(0) = -4. |3 - (-4)| = 7. s(5) = -29. |s(5) - s(1)| = |-29 - 3| = 32. s(7) = 3. |s(7) - s(5)| = |3 - (-29)| = 32. Sum = 7 + 32 + 32 = 71. The arithmetic is actually correct. Let me re-read the prompt's constraint. "Derivative... unchecked". The equations are marked unchecked, but the prompt says "a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)". Wait, the lines are marked `[equation, unchecked]`. This usually implies the CAS hasn't checked them, or the user is asking me to check them? No, the prompt says "Each line... is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE". The tags in the solution say `unchecked`. This is a contradiction in the prompt format vs the solution tags. However, usually, in these tasks, if it says `equation`, I assume the algebra is correct unless it's obviously wrong. Let's look closer at the sentence logic. Step 4 says "v changes sign... so the particle turns around there." This is correct for 1D motion. Step 5 calculates the distance. The values s(0)=-4, s(1)=3, s(5)=-29, s(7)=3 are correct. The distances are 7, 32, 32. Sum 71. The solution seems correct. Why would it fail? Let me check the acceleration sign or something? No, distance only depends on velocity zeros. Is there a subtlety? "Find its velocity and acceleration..." The solution provides them. "times it is at rest" provided. "total distance" provided. The logic holds. The arithmetic holds. The setup holds. I will mark it pass.
gpt-oss:20b: pass 2026-10-10

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/particle_motion, checked 2026-10-10 with SymPy 1.14.0.