10.7: Constancy of Momentum and Isolated Systems

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Suppose we now completely isolate our system from the surroundings. When the external force acting on the system is zero,

\[\overrightarrow{\mathbf{F}}^{\operatorname{ext}}=\overrightarrow{\boldsymbol{0}} \nonumber \]

the system is called an isolated system. For an isolated system, the change in the momentum of the system is zero,

\[\Delta \overrightarrow{\mathbf{p}}_{\mathrm{sys}}=\overrightarrow{\mathbf{0}} \quad \text { (isolated system) } \nonumber \]

therefore the momentum of the isolated system is constant. The initial momentum of our system is the sum of the initial momentum of the individual particles,

\[\overrightarrow{\mathbf{p}}_{\mathrm{sys}, i}=m_{1} \overrightarrow{\mathbf{v}}_{1, i}+m_{2} \overrightarrow{\mathbf{v}}_{2, i}+\cdots \nonumber \]

The final momentum is the sum of the final momentum of the individual particles,

\[\overrightarrow{\mathbf{p}}_{\mathrm{sys}, f}=m_{1} \overrightarrow{\mathbf{v}}_{1, f}+m_{2} \overrightarrow{\mathbf{v}}_{2, f}+\cdots \nonumber \]

Note that the right-hand-sides of Equations. (10.7.3) and (10.7.4) are vector sums.

When the external force on a system is zero, then the initial momentum of the system equals the final momentum of the system,

\[\overrightarrow{\mathbf{p}}_{\mathrm{sys}, i}=\overrightarrow{\mathbf{p}}_{\mathrm{sys}, f} \nonumber \]