Projectile motion
Problem 9.821 · medium
A projectile is fired from \( \displaystyle 1 \) ft above the ground at \( \displaystyle 128 \) 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) = 1 + v₀sin θ · t − ½gt².
- \[ \left. \frac{d}{d t} \left(- 16 t^{2} + 64 \sqrt{3} t + 1\right) \right|_{\substack{ t=2 \sqrt{3} }} = 0 \]The vertical velocity is 0 at t = 2*sqrt(3).✓ Proved
- \[ \left. - 16 t^{2} + 64 \sqrt{3} t + 1 \right|_{\substack{ t=2 \sqrt{3} }} = 193 \]The maximum height.✓ Proved
- \[ \left. - 16 t^{2} + 64 \sqrt{3} t + 1 \right|_{\substack{ t=2 \sqrt{3} + \frac{\sqrt{193}}{4} }} = 0 \]It lands (y = 0) at t = 2*sqrt(3) + sqrt(193)/4.✓ Proved
- \[ \left. 64 t \right|_{\substack{ t=2 \sqrt{3} + \frac{\sqrt{193}}{4} }} = 128 \sqrt{3} + 16 \sqrt{193} \]The horizontal distance then.✓ Proved
Answer \( H = 193 \approx 193.00,\ T = 2 \sqrt{3} + \frac{\sqrt{193}}{4} \approx 6.937,\ \text{range} = 128 \sqrt{3} + 16 \sqrt{193} \approx 443.98 \)
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: fail (error) — The horizontal position function is incorrect; it uses 64t instead of 64\sqrt{3}t, leading to an incorrect range calculation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The horizontal position function is incorrect; it uses 64t instead of 64\sqrt{3}t, leading to an incorrect range calculation.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (error) 2026-10-19 — The horizontal position function is incorrect; it uses 64t instead of 64*sqrt(3)t, leading to an incorrect range calculation.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to explicitly calculate the horizontal velocity component (v₀cos θ = 64√3) used in the x(t) equation, jumping directly to using '64*t' in step 5 without showing the derivation from the given 128 ft/s and 60°. Additionally, the time of flight calculation in step 4 is presented as a verification rather than a derivation, obscuring the quadratic formula application needed to find the positive root.gpt-oss:20b: pass 2026-10-19
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 2026-10-20 with SymPy 1.14.0.