External Impulse and Breakdown of Conservation

When an external impulse breaks conservation and how to estimate whether its effect is negligible.

Conceptual context

Linear momentum conservation is a conditional principle: it holds only if the net external force is zero. When an external impulse acts on the system, the total momentum changes and conservation is broken. Understanding exactly how and how much it breaks is essential for deciding whether a problem admits the approximation of conserved momentum or if it requires using the full form of the impulse balance.

In practice, most real situations involve external forces: friction with the ground, gravity, or string tensions. The key lies in quantifying whether those forces create a significant impulse during the event.

🟢 Essential level

The conservation of a system's motion only works when the system is sufficiently isolated. If something external pushes, brakes, or hits the system for a significant time, the total motion of the set changes.

The essential idea is to distinguish between internal changes and changes produced from the outside. If two parts of the system interact with each other, the change is compensated within the set itself. But if the ground, a wall, an engine, or the air acts, the system receives an external intervention and can no longer be treated as isolated.

[!IMPORTANT] When the outside intervenes, physics has not failed: the external intervention simply needs to be included in the dynamic balance.

Everyday example: a braking car does not lose motion "just because." Friction from the ground acts from outside the car system and reduces its momentum. If the chosen system were car plus Earth, that interaction would no longer be outside; that's why the choice of system changes the physical reading of the problem.

🔵 Formal level

Basic formulas

Leaf Nomenclature

In this leaf: $J$ represents the external impulse; $F_{ext}$ is the net external force; $t$ is the time of action; and $\Delta P$ is the change in linear momentum caused by said impulse.

The Principle Under Ideal Conditions

Before studying the rupture, it is necessary to establish the reference point: the tools that work when the system is perfectly isolated. The necessary and sufficient condition for momentum to be conserved is that the vector sum of all external forces on the system is zero:

Isolation Condition

Under this requirement, the momentum of each individual particle of mass $m$ and velocity $v$ is defined as the product of both quantities:

Definition of Linear Momentum

The total initial momentum of the system is the vector sum of the momenta of all its constituent particles:

Initial Linear Momentum (Sum)

When the system is isolated, conservation establishes equality between the initial and final dynamic states:

Principle of Conservation of Linear Momentum

The net change in momentum under those conditions is exactly zero:

Variation of Linear Momentum

For interactions between two bodies in one dimension, the general balance captures any type of collision —elastic, inelastic, or explosion—, relating initial velocities $v_{1i}$ and $v_{2i}$ (of the first and second body, with masses $m_1$ and $m_2$ respectively) with final velocities $v_{1f}$ and $v_{2f}$:

General 1D Momentum Balance (2 bodies)

If the bodies fuse after the impact (perfectly inelastic collision), the joint final velocity $v_f$ is given by:

Perfectly Inelastic Collision (Coupling)

In the opposite case —when a system at rest separates into two fragments—, the recoil law determines the relative velocities:

Recoil Law (Explosion from rest)

These tools are valid exclusively when the isolation condition is satisfied. As soon as this condition fails, the system enters the regime described below.

Real student questions

• When can I ignore gravity? When the impact is brief (milliseconds). The gravitational impulse is tiny compared to the contact forces.
• What if the impulse is perpendicular to the motion? It changes the system's direction but does not affect the magnitude of the momentum in the original direction.

Final synthesis

Momentum conservation breaks whenever there is a net external impulse. Evaluating the magnitude of this impulse allows for deciding whether the conservation approximation is admissible or if a full dynamic analysis is necessary.