System with external forces — conditions for conservation

WHAT we are talking about

The key conceptual move: conservation depends on where you draw the boundary. Gravity is "external" if only a ball is your system, but "internal" if Earth + ball is your system.

HOW: derive the condition from Newton's laws (from scratch)

Step 1 — Newton's 2nd law for one particle. $\vec F_i = \frac{d\vec p_i}{dt}$ Why this step? This is the definition of force in momentum form — our only starting axiom.

Step 2 — Split the force on particle $i$. Every force on $i$ is either from another member $j$ (internal) or from outside (external): $\vec F_i = \underbrace{\sum_{j\neq i}\vec f_{ij}}_{\text{internal}} + \underbrace{\vec F_i^{ext}}_{\text{external}}$ Why? Every force has a source; that source is either in or out of our chosen system. No third option.

Step 3 — Sum over all particles. $\sum_i \vec F_i = \sum_i\sum_{j\neq i}\vec f_{ij} + \sum_i \vec F_i^{ext} = \frac{d}{dt}\sum_i \vec p_i = \frac{d\vec P}{dt}$

Step 4 — Kill the internal sum using Newton's 3rd law. For each pair, $\vec f_{ij} = -\vec f_{ji}$, so they cancel in pairs: $\sum_i\sum_{j\neq i}\vec f_{ij} = \vec 0$ Why? Internal forces come in equal-and-opposite twins; their vector sum is zero. This is why internal forces can never change total momentum (think of an explosion: fragments fly out, but $\vec P$ of all pieces is unchanged).

Step 5 — The master equation. $\boxed{\dfrac{d\vec P}{dt} = \sum_i \vec F_i^{ext} = \vec F_{ext}^{net}}$

WHY each relaxation is true

Worked examples

Common mistakes (steel-manned)

Connections

• Newton's Third Law — why internal forces cancel.
• Impulse–Momentum Theorem — $\Delta \vec P = \int \vec F\,dt$, basis of the impulsive approximation.
• Center of Mass Motion — $\vec F_{ext}^{net} = M\vec a_{cm}$, same equation, restated.
• Elastic vs Inelastic Collisions — separate condition for kinetic energy.
• Conservation of Linear Momentum — the special case $\vec F_{ext}=0$.

Concept Map

Hinglish (regional understanding)