KAHIBARO
Table of Contents
What an Isolated System Means
In momentum problems, an isolated system is a collection of objects that does not experience any net external impulse during the time interval we study. In simpler words, objects inside the system may push or pull on each other, but influences from outside the system are absent, or so small that we can ignore them.
The idea of a system is very important. We choose which objects to include. Once the boundary of the system is chosen, every force is classified as either internal or external. Internal forces act between objects inside the system. External forces come from outside the chosen system.
If the net external force is zero, or if its effect over the time interval is negligible, the total momentum of the system remains constant.
For an isolated system,
$$\sum \vec{F}_{\text{ext}} = \vec{0}$$
and therefore
$$\frac{d\vec{P}}{dt} = \vec{0}$$
so the total momentum is constant:
$$\vec{P}_{\text{initial}} = \vec{P}_{\text{final}}$$
where
$$\vec{P} = \sum_i m_i \vec{v}_i$$
Internal and External Forces
Suppose two skaters push each other on ice. If we choose both skaters as the system, the push between them is internal. Their individual momenta change, but the total momentum of the two-skater system stays the same if friction is negligible.
If we choose only one skater as the system, then the push from the other skater becomes an external force. In that case, the momentum of that one-skater system changes.
This shows that isolation is not only about the physical situation, but also about how we define the system.
Why Internal Forces Do Not Change Total Momentum
Internal forces come in action reaction pairs. One object pushes another, and the second pushes back with equal magnitude and opposite direction. These internal forces may change the momenta of individual parts, but when we add the momentum of all parts together, the internal effects cancel.
For two objects inside a system,
$$\vec{F}_{12} = -\vec{F}_{21}$$
Because momentum changes according to force, the total change due to these internal forces cancels when we sum over the whole system.
This is the reason momentum conservation is so powerful in collisions, explosions, recoil, and many-body motion.
Isolated Versus Non-Isolated Systems
In real life, perfectly isolated systems are rare. But many systems are isolated approximately for a short time. A collision between billiard balls is a good example. Gravity and friction may exist, but during the very short collision time, their impulses are much smaller than the collision forces. So the system can often be treated as isolated.
The key idea is not always that every external force is exactly zero, but that the net external impulse is negligible.
A system can be treated as isolated over a time interval if the net external impulse is negligible:
$$\vec{J}_{\text{ext}} = \int \sum \vec{F}_{\text{ext}} \, dt \approx \vec{0}$$
Then,
$$\Delta \vec{P} \approx \vec{0}$$
Examples of Isolated Systems
The table below shows common situations and whether the system can be treated as isolated.
| Situation | Chosen system | Isolated? | Reason |
|---|---|---|---|
| Two carts colliding on a nearly frictionless track | Both carts | Approximately yes | External friction is very small |
| Gun firing a bullet | Gun and bullet | Approximately yes during firing | Internal forces dominate, external impulse is small during short interval |
| Rocket in space ejecting gas | Rocket plus expelled gas | Yes, if outside forces are negligible | Thrust is internal to the chosen system |
| Falling ball near Earth | Ball only | No | Gravity is an external force on the ball |
| Falling ball plus Earth | Ball and Earth | Much closer to yes | Gravitational interaction becomes internal |
Choosing the Right System
A smart choice of system often makes a problem much easier. If two objects interact strongly with each other, it is often useful to include both in the same system. Then their interaction forces become internal, and momentum conservation may apply.
For example, in recoil problems, choosing the recoiling object and the emitted object together usually gives a conserved total momentum. Before interaction, the system may be at rest, so total momentum is zero. After interaction, the momenta must still add to zero.
If initially the system is at rest,
$$\vec{P}_{\text{initial}} = \vec{0}$$
Then after separation,
$$\sum_i \vec{p}_i = \vec{0}$$
For two objects,
$$m_1 \vec{v}_1 + m_2 \vec{v}_2 = \vec{0}$$
This means their momenta are equal in magnitude and opposite in direction.
Time Interval Matters
Whether a system is isolated can depend on the time interval we consider. A football in flight is not an isolated system over a long time, because gravity continuously changes its momentum. But during a very brief collision with another object, the external impulse from gravity may be tiny compared with the internal collision impulse. Over that short collision time, the two-object system can be treated as isolated.
So when applying momentum conservation, always ask, "Over what time interval am I studying the system?"
A Visual Picture
In this drawing, the forces between $m_1$ and $m_2$ are internal because both objects are inside the boundary. The red force is external because it comes from outside the system.
Connection to Total Momentum
The total momentum of a system is the vector sum of the momenta of all its parts:
$$\vec{P} = \vec{p}_1 + \vec{p}_2 + \vec{p}_3 + \cdots$$
If the system is isolated, this total vector does not change. Individual momenta can change a lot, but the sum remains the same.
This is especially important because momentum has direction. In one dimension, signs handle direction. In two and three dimensions, each component of total momentum is conserved separately for an isolated system.
In an isolated system, momentum conservation applies component by component:
$$P_{x,\text{initial}} = P_{x,\text{final}}$$
$$P_{y,\text{initial}} = P_{y,\text{final}}$$
$$P_{z,\text{initial}} = P_{z,\text{final}}$$
Common Mistakes
A frequent mistake is to say that if forces exist, momentum cannot be conserved. This is false. Internal forces can be large, and momentum can still be conserved for the whole system.
Another common mistake is to apply conservation of momentum to a single object when strong external forces act on it. In that case, the object alone is not an isolated system.
A third mistake is forgetting the time interval. A system may be approximately isolated during a short interaction even if it is not isolated over a long period.
Summary
An isolated system is a system for which the net external force, or more precisely the net external impulse over the time interval considered, is zero or negligible. Internal forces may change the motion of individual parts, but they do not change the total momentum of the system. When a system is isolated, its total momentum remains constant, which is the foundation of momentum conservation in mechanics.