Projectile motion
Problem 9.673 · medium
A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 32 \) ft/s, \( \displaystyle 60^\circ \) above the horizontal. Taking \( \displaystyle g = 32 \) ft/s², find the maximum height, the time of flight and the horizontal range.
- x(t) = v₀cos θ · t and y(t) = 0 + v₀sin θ · t − ½gt².
- \[ \left. \frac{d}{d t} \left(- 16 t^{2} + 16 \sqrt{3} t\right) \right|_{\substack{ t=\frac{\sqrt{3}}{2} }} = 0 \]The vertical velocity is 0 at t = sqrt(3)/2.✓ Proved
- \[ \left. - 16 t^{2} + 16 \sqrt{3} t \right|_{\substack{ t=\frac{\sqrt{3}}{2} }} = 12 \]The maximum height.✓ Proved
- \[ \left. - 16 t^{2} + 16 \sqrt{3} t \right|_{\substack{ t=\sqrt{3} }} = 0 \]It lands (y = 0) at t = sqrt(3).✓ Proved
- \[ \left. 16 t \right|_{\substack{ t=\sqrt{3} }} = 16 \sqrt{3} \]The horizontal distance then.✓ Proved
Answer \( H = 12 \approx 12.00,\ T = \sqrt{3} \approx 1.732,\ \text{range} = 16 \sqrt{3} \approx 27.71 \)
Lines: 4 proved, 1 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | the flight stepped forward in time steps of 10⁻⁴ s |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The horizontal position function x(t) is incorrectly defined as 16t (implying v_x = 16 ft/s) in lines 5 and the final answer, whereas v_x = 32 cos(6
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The horizontal position function x(t) is incorrectly defined as 16t (implying v_x = 16 ft/s) in lines 5 and the final answer, whereas v_x = 32 cos(6gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The horizontal position function x(t) is incorrectly defined as 16t in line 5 (and implied in line 1's application), whereas v₀cos(60°) = 16 ft/s, sgpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The horizontal velocity component is calculated incorrectly as 16 ft/s instead of 16\sqrt{3} ft/s (v_0 \cos(60^\circ) = 32 \cdot 0.5 = 16 is correct, wait. 32 * cos(60) = 16. 32 * sin(60) = 16\sqrt{3}. The vertical equation uses 16\sqrt{3}t, which is correct. The horizontal equation in step 5 uses 16t. Let's re-read step 1. Step 1 says x(t) = v_0 cos(theta) t. v_0=32, theta=60. cos(60)=1/2. So x(t) = 16t. This is correct. Step 5 substitutes t=sqrt(3) into 16t to get 16\sqrt{3}. This is correct. The stated answer says range = 16\sqrt{3}. This is correct. Why did I think it was wrong? Let me re-evaluate. v_x = 32 cos(60) = 16. v_y = 32 sin(60) = 16\sqrt{3}. y(t) = 16\sqrt{3}t - 16t^2. Max height at t = v_y/g = 16\sqrt{3}/32 = \sqrt{3}/2. H = 16\sqrt{3}(\sqrt{3}/2) - 16(\sqrt{3}/2)^2 = 24 - 16(3/4) = 24 - 12 = 12. Correct. Time of flight: y(t)=0 => t(16\sqrt{3} - 16t) = 0 => t = \sqrt{3}. Correct. Range = v_x * T = 16 * \sqrt{3}. Correct. The solution steps are algebraically correct and match the stated answer. The sentences are minimal but correct. Step 1 sets up the model correctly. Steps 2-5 perform the calculations. There are no errors.gpt-oss:20b: pass 2026-10-14
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/projectile_motion, checked 2026-10-17 with SymPy 1.14.0.