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Projectile motion

Problem 9.823 · medium

A projectile is fired from \( \displaystyle 2 \) ft above the ground at \( \displaystyle 128 \) 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
\[ \left. \frac{d}{d t} \left(- 16 t^{2} + 64 t + 2\right) \right|_{\substack{ t=2 }} = 0 \]
The vertical velocity is 0 at t = 2.✓ Proved
\[ \left. - 16 t^{2} + 64 t + 2 \right|_{\substack{ t=2 }} = 66 \]
The maximum height.✓ Proved
\[ \left. - 16 t^{2} + 64 t + 2 \right|_{\substack{ t=2 + \frac{\sqrt{66}}{4} }} = 0 \]
It lands (y = 0) at t = 2 + sqrt(66)/4.✓ Proved
\[ \left. 64 \sqrt{3} t \right|_{\substack{ t=2 + \frac{\sqrt{66}}{4} }} = 16 \sqrt{3} \left(8 + \sqrt{66}\right) \]
The horizontal distance then.✓ Proved

Answer \( H = 66 \approx 66.00,\ T = 2 + \frac{\sqrt{66}}{4} \approx 4.031,\ \text{range} = 16 \sqrt{3} \left(8 + \sqrt{66}\right) \approx 446.84 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly models the projectile motion with the given initial conditions and derives the correct expressions for maximum height, time of flight, and range.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly models the projectile motion with the given initial conditions and derives the correct expressions for maximum height, time of flight, and range.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up the parametric equations for projectile motion with the given initial conditions. The algebraic steps for finding maximum height, time of flight, and range are correct and verified by the CAS checks.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly models the projectile motion with the given initial conditions and accurately computes the maximum height, time of flight, and horizontal range.
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.