Projectile motion
Problem 9.760 · medium
A projectile is fired from \( \displaystyle 2 \) ft above the ground at \( \displaystyle 64 \) ft/s, \( \displaystyle 30^\circ \) 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) = 2 + v₀sin θ · t − ½gt².Reviewed
- \[ \left. \frac{d}{d t} \left(- 16 t^{2} + 32 t + 2\right) \right|_{\substack{ t=1 }} = 0 \]The vertical velocity is 0 at t = 1.✓ Proved
- \[ \left. - 16 t^{2} + 32 t + 2 \right|_{\substack{ t=1 }} = 18 \]The maximum height.✓ Proved
- \[ \left. - 16 t^{2} + 32 t + 2 \right|_{\substack{ t=1 + \frac{3 \sqrt{2}}{4} }} = 0 \]It lands (y = 0) at t = 1 + 3*sqrt(2)/4.✓ Proved
- \[ \left. 32 \sqrt{3} t \right|_{\substack{ t=1 + \frac{3 \sqrt{2}}{4} }} = 32 \sqrt{3} + 24 \sqrt{6} \]The horizontal distance then.✓ Proved
Answer \( H = 18 \approx 18.00,\ T = 1 + \frac{3 \sqrt{2}}{4} \approx 2.061,\ \text{range} = 32 \sqrt{3} + 24 \sqrt{6} \approx 114.21 \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly sets up the parametric equations for projectile motion with the given initial conditions. The subsequent algebraic steps for finding maximum height, time of flight, and range are verified as correct by the provided equation checks.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly sets up the parametric equations for projectile motion with the given initial conditions. The subsequent algebraic steps for finding maximum height, time of flight, and range are verified as correct by the provided equation checks.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly sets up the parametric equations for projectile motion with the given initial conditions. The derived values for maximum height, time of flight, and horizontal range are algebraically correct and match the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the parametric equations for projectile motion with the given initial conditions. The subsequent algebraic steps for finding maximum height, time of flight, and range are correct and consistent with the stated answer.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.