Isolated Systems

System model with negligible external impulse, essential for conserving total momentum.

Conceptual context

An isolated system is one on which the net external force $J_{ext}$ is zero. This does not mean there are no forces: particles inside the system can interact intensely with each other (internal forces), but the sum of all forces arriving from outside cancels out. In that situation, the total linear momentum of the system is conserved exactly.

When does this happen in practice? Strictly speaking, never — gravity, air friction, etc. are always present. But in many scenarios the system is approximately isolated during the event of interest: a collision lasting milliseconds, an internal explosion, or a system in outer space where gravity is negligible. Treating the system as isolated is then an excellent approximation.

🟢 Essential level

The essential idea of an isolated system is a modeling decision: the goal is to determine whether the chosen set of bodies can be treated as if the environment does not significantly alter its overall motion during the time interval under study. When this is the case, the system’s behavior is governed by what happens internally rather than by external influences.

This does not imply the absence of forces. Within the system, there may be strong contacts, tensions, or even explosive interactions; what matters is that these effects compensate when the system is considered as a whole. This is why the system boundary is just as important as the final calculation.

Two bodies colliding on a table may not be isolated in the vertical direction and yet behave as an isolated system horizontally during the brief impact. The key question is always: in which component and over what interval does the environment cease to dominate?

🔵 Formal level

Basic formulas

At the formal level, the isolated system is defined by evaluating the vector sum of all forces acting on each particle. According to Newton's Third Law, internal forces form action-reaction pairs that cancel out when summed over the entire system.

This implies that the temporal rate of change of the total momentum depends exclusively on the net external force. If this external force is zero, the total momentum remains strictly constant.

To build this balance, we first define the initial total momentum $p_{tot,i}$ of the assembly:

Initial total momentum

And the final total momentum $p_{tot,f}$ of the particles after the interaction:

Final total momentum

When the isolation is perfect or the interaction is sufficiently brief, we establish conservation by equating both states:

Linear momentum conservation

Real student questions

• Does a perfectly isolated system exist? Strictly, no. But the approximation is excellent when external forces are much weaker than internal ones, or act perpendicular to the direction of interest.
• Does gravity prevent a system from being isolated? It depends. If motion is horizontal and the normal balances gravity vertically, the system can be isolated in the horizontal direction.
• What if I chose the wrong system boundaries? Total momentum will not be conserved, but you can correct by using the impulse theorem to incorporate the effect of the external force.

Final synthesis

A system is isolated when the net external force is zero, making its total momentum exactly conserved. In practice, the "isolation" decision is always an approximation governed by three factors: boundary definition, time scale, and direction. Mastering this decision is the key to correctly applying momentum conservation in any collision or explosion problem.