Can Internal Forces Change a System’s Total Momentum?

Internal forces can make parts of a system speed up, slow down, or move in opposite directions. But in Newtonian mechanics, they do not change the total momentum of an isolated system. The reason is that internal forces occur in equal-and-opposite pairs, so their impulses cancel when you add the momentum changes of all parts.

The system’s total momentum changes if it experiences a net external impulse. The key is to define the system boundary carefully: a force is internal only if both interacting objects are included.

Total momentum and force

For objects with masses $m_i$ and velocities $\mathbf{v}_i$, the total momentum is the vector sum

$$\mathbf{P}=\sum_i m_i\mathbf{v}_i.$$

For a fixed collection of objects in Newtonian mechanics, the rate of change of total momentum equals the net external force:

$$\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{external, net}}.$$

To see why internal forces drop out, consider two objects that exert forces on one another. Newton’s third law gives $\mathbf{F}_{12}=-\mathbf{F}_{21}$. Adding their equations of motion cancels this pair:

$$\frac{d}{dt}(\mathbf{p}_1+\mathbf{p}_2) =\mathbf{F}_{12}+\mathbf{F}_{21}+\mathbf{F}_{\text{external}} =\mathbf{F}_{\text{external}}.$$

The same cancellation applies when you sum over all objects in the system, provided the internal forces pair up this way. If the net external force is zero, total momentum stays constant. More generally, the change in total momentum is the external impulse:

$$\Delta\mathbf{P}=\int_{t_1}^{t_2}\mathbf{F}_{\text{external, net}}\,dt.$$

Worked example: two skaters push apart

Suppose two skaters are initially at rest on nearly frictionless ice. One has mass $60\,\text{kg}$ and the other has mass $40\,\text{kg}$. They push off each other. During the push, the forces are internal to the two-skater system and are equal and opposite.

If the first skater ends up moving at $-2\,\text{m/s}$, their momentum is

$$p_1=(60\,\text{kg})(-2\,\text{m/s})=-120\,\text{kg}\cdot\text{m/s}.$$

The system started with zero total momentum, so the second skater must have momentum $+120\,\text{kg}\cdot\text{m/s}$. Their velocity is therefore

$$v_2=\frac{120\,\text{kg}\cdot\text{m/s}}{40\,\text{kg}}=3\,\text{m/s}.$$

The skaters have gained momentum in opposite directions, but the total remains zero:

$$P=p_1+p_2=-120+120=0.$$

The internal push changed each skater’s momentum, not the momentum of the pair as a whole. In a real experiment, friction or another outside influence may exert a small external impulse, so the total may not be exactly constant.

When internal forces seem to change momentum

Usually, an apparent exception comes from the system boundary or from leaving out something that carries momentum.

• The system is only part of the interaction. If you include a ball but not the person throwing it, the person’s force on the ball is external to the ball-only system. The ball’s momentum can change.
• Matter crosses the boundary. A rocket is not a fixed collection of the same material as it burns fuel. To track momentum correctly, account for the momentum carried out by the exhaust, or include the rocket and exhaust together as a larger system.
• A field carries momentum. In electromagnetic interactions, momentum can be transferred between matter and the electromagnetic field. If you track only the matter, its momentum may change even though the total momentum of matter plus field is conserved.

These cases do not mean that internal forces change the total momentum of a properly defined, isolated system. They mean that the chosen system is open, or that a momentum-carrying part was left out. In introductory Newtonian problems, the usual assumption is that the system is a fixed set of objects and internal force pairs obey Newton’s third law.

Common mistakes

Confusing individual momentum with total momentum. An internal collision can change the momentum of each object, even while the vector sum stays constant. Momentum is a vector, so opposite directions have opposite signs.

Assuming internal forces do nothing. They can greatly change the motion inside the system. An explosion, for example, can send pieces flying in different directions while leaving the total momentum unchanged.

Ignoring the system boundary. A force is not automatically internal because it occurs during an interaction. It is internal only when both the force’s source and the object it acts on are inside the system.

Mixing up momentum and kinetic energy. Internal forces can change a system’s total kinetic energy. A spring releasing stored energy can make parts move faster, even when total momentum is conserved. Momentum conservation alone does not imply kinetic-energy conservation.