Velocity changes are caused by interactions across a system boundary. This page focuses on how to connect net external force to changes in a system’s center-of-mass velocity, using system-based free-body reasoning.

Core idea: only external forces can change a system’s velocity

When you treat one object or many objects as a single system, the motion you track is the motion of the system’s center of mass. Internal pushes and pulls can change how parts move relative to each other, but they cannot change the system’s overall velocity by themselves.

Key terms for system thinking

Choose the system boundary first, then decide which forces cross it (external) and which occur between objects inside it (internal).

A useful velocity to track is the system’s center-of-mass velocity.

Vector construction for the center of mass of several particles: each position vector $\vec r_i$ is weighted by mass ($m_i\vec r_i$), summed, and then divided by total mass to locate $\vec r_{\text{cm}}$. This makes the “mass-weighted average position” idea concrete, which is the basis for discussing the corresponding center-of-mass velocity. Source

Identifying the net external force

The net external force is the vector sum of all external forces on the system.

A system-level free-body diagram: the entire system is represented by a single point (the center of mass), and only external forces are drawn acting on it. The vectors can then be added to obtain the net external force, which determines the direction of the system’s acceleration ($\sum F_{\text{ext}} = M a_{\text{cm}}$). Source

Practically, that means:

• Draw a free-body diagram for the entire system (not for each object).

• Include only forces exerted by the environment on the system.

• Exclude internal forces (forces between objects inside the system), because they occur in action-reaction pairs within the boundary.

Why internal forces do not change the system’s velocity

Internal forces can be large (e.g., tension between two blocks in a system), but they appear in equal and opposite pairs on different parts of the system. When you add forces for the system as a whole, those internal pairs cancel, leaving only external forces to determine whether the system’s center-of-mass velocity changes.

Linking net external force to changing velocity

If the net external force is zero, the center-of-mass velocity is constant (it may be zero or nonzero).

If the net external force is not zero, the center-of-mass velocity must change in the direction of that net external force.

The quantitative link is Newton’s second law applied to a system:

This equation is a direct statement of the syllabus focus: a change in $v_{\text{cm}}$ requires $a_{\text{cm}} \neq 0$, which requires $\sum F_{\text{ext}} \neq 0$.

Direction and components

Because forces are vectors:

• A nonzero net external force in one direction changes the center-of-mass velocity component in that direction.

• It is possible for forces to balance in one axis (no change in that velocity component) while remaining unbalanced in another axis (velocity changes there).

Common modelling moves and pitfalls

Choosing a good system

Pick a boundary that makes the external forces simple.

• If two objects pull on each other with tension, including both objects often removes tension from the external-force list.

• If friction from the floor matters, the floor is usually outside the system, so friction is external (unless you intentionally include the floor too).

Interpreting “no net external force”

“No net external force” does not mean “no forces.” It means:

• External forces can be present but add to zero as vectors.

• With $\sum F_{\text{ext}} = 0$, the system’s center-of-mass velocity stays constant, even if internal forces cause spinning, stretching, or separation of parts.