Source: https://guthlabs.ai/docs/instruments/optimal-launch-angle-finder/

instrument / verified-clean

# Optimal Launch Angle Finder

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Inspectable records

[01**Overview**Purpose and lifecycle](<https://guthlabs.ai/docs/instruments/optimal-launch-angle-finder/>)[02**Mechanism**Method and assumptions](<https://guthlabs.ai/docs/instruments/optimal-launch-angle-finder/mechanism/>)[03**Implementation**Binding and validation](<https://guthlabs.ai/docs/instruments/optimal-launch-angle-finder/implementation/>)[04**Qualification**Evidence and errata](<https://guthlabs.ai/docs/instruments/optimal-launch-angle-finder/qualification/>)

## Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

**Area**: Physics

**Tranche**: tranche-2

**Evidence ID**: t2-023

**Ledger rows**: 1

No ledger note is attached.

## Review findings

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.

### Formula and scope findings

- 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.

### Unit assumptions

- 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.

## Boundary hardening

**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.

## Errata and corrections

No applied erratum is attached to this unique-name record.

## Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

````
{
  "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."
}
````

## Related workflows

[Kinematics & Collisions Tutorrange optimization](<https://guthlabs.ai/docs/instrument-workflows/kinematics-tutor/>)

## Sources and evidence

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

- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`

- `evidence-tranche2/tranche2-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`