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Projectile motion

Problem 9.671 · medium

A projectile is fired from \( \displaystyle 0 \) ft above the ground at \( \displaystyle 96 \) 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) = 0 + v₀sin θ · t − ½gt².
Reviewed
\[ \left. \frac{d}{d t} \left(- 16 t^{2} + 48 t\right) \right|_{\substack{ t=\frac{3}{2} }} = 0 \]
The vertical velocity is 0 at t = 3/2.✓ Proved
\[ \left. - 16 t^{2} + 48 t \right|_{\substack{ t=\frac{3}{2} }} = 36 \]
The maximum height.✓ Proved
\[ \left. - 16 t^{2} + 48 t \right|_{\substack{ t=3 }} = 0 \]
It lands (y = 0) at t = 3.✓ Proved
\[ \left. 48 \sqrt{3} t \right|_{\substack{ t=3 }} = 144 \sqrt{3} \]
The horizontal distance then.✓ Proved

Answer \( H = 36 \approx 36.00,\ T = 3 \approx 3.000,\ \text{range} = 144 \sqrt{3} \approx 249.42 \)

✓ 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 sets up the kinematic equations, identifies the time of maximum height and landing time, and computes the requested quantities accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly sets up the kinematic equations, identifies the time of maximum height and landing time, and computes the requested quantities accurately.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the kinematic equations, identifies the time of maximum height and landing time, and computes the correct values for height, time of flight, and range.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the kinematic equations, identifies the time to maximum height and landing time, and calculates the corresponding height and range accurately.
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.