Documentation / @interstellar-tools/equations / combineBurnsDeltaV

Function: combineBurnsDeltaV()

function combineBurnsDeltaV(
  v1: number,
  v2: number,
  deltaTheta: Radians
): number;

Defined in: manoeuvres/combine-burns-delta-v.ts:71

Combine non-collinear burns (vector law / law of cosines for velocities).

Definition

$$\mathrm{\Delta }v=\sqrt{v_1^2+v_2^2-2v_1{v}_2\cos \mathrm{\Delta }\theta }$$

where:

• ($v_1$) and ($v_2$) are the magnitudes of two impulsive burns, and
• ($\mathrm{\Delta }\theta$) is the angle between their directions (radians).

Units

• Inputs: ($v_1,v_2$) in m/s, ($\mathrm{\Delta }\theta$) in radians.
• Output: ($\mathrm{\Delta }v$) in m/s.

Domain

• ($v_1\ge 0,v_2\ge 0,\mathrm{\Delta }\theta \in [0,\pi ]$).

Notes

• Special cases:
  • ($\mathrm{\Delta }\theta =0$) (same direction) → ($\mathrm{\Delta }v=|v_1-v_2|$).
  • ($\mathrm{\Delta }\theta =\pi$) (opposite) → ($\mathrm{\Delta }v=v_1+v_2$).
• This computes the single equivalent impulse from two burns applied with mutual angle ($\mathrm{\Delta }\theta$) (i.e., magnitude of the vector sum).

Parameters

Parameter	Type	Description
v1	number	Magnitude of burn 1 (m/s). Must be finite and ≥ 0.
v2	number	Magnitude of burn 2 (m/s). Must be finite and ≥ 0.
deltaTheta	Radians	Angle between burn directions (radians), in [0, π].

Returns

number

Equivalent single ($\mathrm{\Delta }v$) (m/s).

Throws

If any input is not a number.

Throws

If inputs are non-finite or out of domain.

Example

import { combineBurnsDeltaV } from '@interstellar-tools/equations';
import type { Radians } from '@interstellar-tools/types';

// Helper: degrees → branded radians
const toRadians = (deg: number): Radians => ((deg * Math.PI) / 180) as Radians;

// Two burns (magnitudes in m/s)
const v1 = 500; // first impulse
const v2 = 300; // second impulse

// 1) Same direction (Δθ = 0°) → Δv = |v1 - v2|
const dv_same = combineBurnsDeltaV(v1, v2, toRadians(0));
console.log('Δθ = 0°   → Δv =', dv_same.toFixed(3), 'm/s');

// 2) Opposite directions (Δθ = 180°) → Δv = v1 + v2
const dv_opp = combineBurnsDeltaV(v1, v2, toRadians(180));
console.log('Δθ = 180° → Δv =', dv_opp.toFixed(3), 'm/s');

// 3) General non-collinear case (e.g., Δθ = 60°)
const dv_60 = combineBurnsDeltaV(v1, v2, toRadians(60));
console.log('Δθ = 60°  → Δv =', dv_60.toFixed(3), 'm/s');

// 4) Another example (Δθ = 25°)
const dv_25 = combineBurnsDeltaV(1200, 800, toRadians(25));
console.log('Δθ = 25°  → Δv =', dv_25.toFixed(3), 'm/s');