Impulse theorem and the fence diagram

For a system of bodies, the change in the total momentum equals the sum of the impulses of the external forces only:

Law — Impulse theorem

$\vv{p}_{\text{tot}}^{\,\text{fin}} - \vv{p}_{\text{tot}}^{\,\text{ini}} = \sum \vv{I}_{\text{esterne}}$

The reason only external forces count is Newton’s third law: the impulses of the internal forces cancel in pairs, because every action inside the system has its equal and opposite reaction, itself also inside the system.

The fence diagram is the momentum analogue of the interaction diagram for forces and the bubble diagram for energy. The bodies are drawn as diamonds or circles inside a fence (the system boundary), and the impulses as arrows. The internal arrows — between bodies both inside the fence — cancel out; only arrows that cross the boundary change $\vv{p}_\text{tot}$.

Fence diagram: internal impulses cancel; only those crossing the boundary change $\vv{p}_\text{tot}$. Same logic as the interaction diagram and the bubble diagram.

Why it is the second law in disguise

The impulse theorem is really Newton’s second law rewritten in integral form: $\vv{F} = m\vv{a} = m\frac{\Delta\vv{v}}{\Delta t} = \frac{\Delta\vv{p}}{\Delta t}$ Multiplying by $\Delta t$ gives $\vv{F}\,\Delta t = \Delta\vv{p}$, that is, impulse equals change in momentum.

Links

Topics: Quantità di moto e urti Concepts: Impulso · Quantità di moto · Terza legge di Newton Methods: Diagramma del recinto

Related exercises: La palla che torna · Crash test con airbag · Auto contro camion