Optimal Launch Angle Finder mechanism

Model scope

The supplied maximum-range angle formula is valid only for the stated vacuum maximum-range problem, not for the broader target-hitting claim. The instrument should either remove target-solving claims or add the projectile trajectory and target-angle equations, feasibility checks, and the complete range formula.

• The level-ground optimum θ_max=45° is correct only for a vacuum projectile with equal launch and landing elevations, positive launch speed, and uniform positive gravitational acceleration.
• The height-offset formula θ=atan(v0/√(v0²+2gh0)) is correct for maximizing horizontal range when h0 is the launch height above the landing level, with v0>0, g>0, and h0≥0. These domain constraints are not stated.
• The specification claims the instrument can find an angle that hits a target, but no target-trajectory equation or target-angle solution is provided. Hitting a target generally yields zero, one, or two feasible angles depending on distance, height, speed, and constraints.
• The specification claims to output resulting range, but no range formula is provided. The required vacuum-model range formula is R=(v0 cosθ/g)(v0 sinθ+√(v0²sin²θ+2gh0)); at the maximizing angle for h0≥0 this simplifies to R_max=(v0/g)√(v0²+2gh0).
• The formula does not define behavior for h0<0, v0=0, g≤0, or v0²+2gh0≤0.

Units and conventions

• Use a consistent unit system: v0 in m/s, g in m/s², h0 and range in m, and angles in degrees for display.
• The example with v0=72 km/h requires conversion to 20 m/s before applying the formula.
• g is uniform gravitational acceleration and is normally 9.81 m/s² near Earth.
• atan returns an angle in radians internally; convert to degrees for the displayed angle.
• h0 means launch height above the landing level, not an arbitrary signed height offset.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "v0": 20,
    "g": 9.81,
    "h0": 0,
    "units": "SI"
  },
  "expected_outputs": {
    "optimal_angle_deg": 45,
    "resulting_range_m": 40.7747
  },
  "working": "θ=atan(20/√(20²+2×9.81×0))=atan(20/20)=atan(1)=45°. R_max=(20/9.81)×√(20²+0)=(20/9.81)×20=400/9.81=40.7747 m."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The height h0=-v0^2/(2g) makes sqrt(v0^2+2gh0)=0, so the angle formula divides by zero and the expected result is undefined. The instrument should reject this input.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche2/tranche2-results.json
• BOUNDARY_HAZARD_CATALOG.json