Impulse-Momentum Theorem for a Mechanical System

The linear momentum of a mechanical system is the vector sum of the momenta of all its particles. The impulse-momentum theorem describes the translational aspect of system motion using only external forces.

Momentum of a system

For a system of $n$ particles:

$$\vec Q=\sum_{i=1}^{n}m_i\vec v_i.$$

Since $\vec v_C=(1/M)\sum m_i\vec v_i$, an important relation follows:

$$\vec Q=M\vec v_C.$$

Differential form

Summing the equations of motion of all particles and using cancellation of internal forces gives:

$$\frac{d\vec Q}{dt}=\sum\vec F^{e}=\vec R^{e}.$$

The rate of change of system momentum equals the resultant external force.

Integral form

Over the interval from $t_1$ to $t_2$:

$$\vec Q_2-\vec Q_1=\int_{t_1}^{t_2}\sum\vec F^{e}\,dt.$$

Thus, the change in total momentum equals the total impulse of the external forces.

Conservation of momentum

If $\sum\vec F^{e}=0$, then $\vec Q=const$. If only the resultant external-force component along one axis is zero, the corresponding momentum component is conserved.

Internal forces

Internal forces may change the velocities of individual parts but do not change total system momentum. This principle is useful in recoil, separation, and many impact problems when external impulse is negligible.

Example

Two bodies of masses 2 kg and 3 kg move along one axis at 4 m/s and -1 m/s. The system momentum is $Q_x=2\cdot4+3\cdot(-1)=5$ kg·m/s. If external impulse along the axis is zero, this value remains constant.

Common mistakes

• adding momentum magnitudes instead of vector components;
• including internal forces in the total external impulse;
• assuming conservation of $\vec Q$ means every particle velocity remains unchanged;
• applying conservation without checking external impulse.