System with external forces — conditions for conservation: Foundations

Before you can trust the parent note's master line $\frac{d P}{d t} = F_{e x t}^{n e t}$ , every letter in it must mean something you can picture. This page builds each symbol from nothing, in an order where each one leans only on the ones before it.

1. Arrows for quantities that have a direction:

Why do we bother? Because a ball moving left at 5 and a ball moving right at 5 are different physical situations, even though both have size 5. A single number cannot tell them apart; an arrow can.

In Figure 1 the red arrow's length is the size (how much) and its tilt is the direction (which way). That is all a vector is — a length with a direction.

Figure 2 shows all three: the tip-to-tail sum, an arrow and its opposite $- b$ , and how they collapse to the zero-length dot $0$ .

2. Mass: $m$ and $M$

Picture: a shopping trolley full of bricks ( $M$ large) versus an empty one ( $m$ small). The same shove moves the empty one much faster. The topic needs $m$ because momentum is mass times velocity — the "how much stuff" is half of the story.

3. Velocity: $v$

Why a vector and not just speed? Because in a collision two objects can have equal speeds $v$ but head-on-opposite velocities $v$ , and the whole outcome depends on that. Speed alone would throw away exactly the information we need.

4. Momentum: $p$ and the total $P$

Figure 3 shows a small light object and a big heavy one with the same momentum arrow: the heavy one moves slower but has more mass, and the arrows match. This is the balance $p = m v$ makes visible.

The subscript $i$ is a counter: $m_{i} v_{i}$ means "the mass and velocity of the $i$ -th object." So $\sum_{i} m_{i} v_{i} = m_{1} v_{1} + m_{2} v_{2} + \dots$ . That is the entire notation — nothing more.

5. Force: $F$ (and the internal / external split)

The double subscript $f_{ij}$ reads "the force on $i$ from $j$ ." So $f_{12}$ is the force on object 1 from object 2, and $f_{21}$ is its twin, on object 2 from object 1.

In Figure 4 the dashed loop is the system boundary — the line you draw to decide who is "in." Arrows crossing the loop are external (red); arrows living entirely inside are internal (black pairs). Everything in this topic hinges on where you draw that loop.

Intuition Why the internal pairs sum to

$0$ (sketch) Take any pair of members $i$ and $j$ . The push on $i$ from $j$ is $f_{ij}$ , and the push on $j$ from $i$ is $f_{j i}$ . These two are equal in size but opposite in direction — pushing apart or pulling together, always symmetric (this equal-and-opposite twinning is Newton's Third Law). In the language of Section 1, that is exactly $f_{j i} = - f_{ij}$ , so the twin adds to nothing: $f_{ij} + f_{j i} = f_{ij} + (- f_{ij}) = 0 .$ Every internal force is one half of such a pair, and when you add the whole system's internal forces they cancel pair by pair: $\sum_{i} \sum_{j \neq = i} f_{ij} = 0 .$ This is why internal pushes — however violent, like an explosion — never budge the total $P$ : their arrows always come with an equal-and-opposite twin that erases them.

6. Rate of change: the derivative $\frac{d}{d t}$

This is the tool the parent note starts from. Nothing deeper is assumed — see Impulse–Momentum Theorem for the reverse view (adding up force over time to get the total change).

7. Putting it together: reading the master equation

Now every symbol in $\frac{d P}{d t} = F_{e x t}^{n e t}$ is earned: $P$ (Section 4) is the group's total momentum arrow; $\frac{d}{d t}$ (Section 6) asks how fast it changes; and $F_{e x t}^{n e t} = \sum_{i} F_{i}^{e x t}$ (Section 5) is the summed-up outside push, after the internal ones have cancelled in pairs. The whole line says: "the group's total momentum changes only as fast as the outside pushes it." If that net outside push is $0$ , the total does not change at all — that is conservation.

Recall Quick self-check on notation

What does the arrow on $p$ tell you that plain $p$ does not? ::: Its direction, not just its size. What does $∣ v ∣$ (or plain $v$ ) mean, and its everyday name? ::: The size/length of the velocity arrow with direction discarded — its speed. In $f_{ij}$ , which object feels the force? ::: Object $i$ (the first subscript) feels it, coming from object $j$ . What does $\sum_{i}$ instruct you to do? ::: Add one term for every object in the system. Write $F_{e x t}^{n e t}$ as a sum. ::: $F_{e x t}^{n e t} = \sum_{i} F_{i}^{e x t}$ — every external push on every member, added as arrows. Why does $\sum_{i} \sum_{j \neq = i} f_{ij} = 0$ ? ::: Internal forces come in equal-and-opposite twins $f_{j i} = - f_{ij}$ , which cancel pair by pair.

Prerequisite map

The diagram below traces the build order used on this page: read it bottom-up. Arrows (Section 1) and mass (Section 2) combine into velocity and then momentum (Sections 3–4); those add up to the total $P$ . Separately, the force split (Section 5) and the derivative (Section 6) give Newton's 2nd law. The total momentum, the internal/external split, and Newton's 2nd law all feed the master equation, which is the parent topic on conservation. (If the diagram does not render in your reader, the same dependency order is spelled out in this paragraph.)

See also Newton's Third Law (why internal pairs cancel) and Center of Mass Motion (the same master equation rewritten).

Equipment checklist

Cover the right side and see if you can state each one before revealing.

A vector is... ::: a quantity with both a size (length) and a direction (tilt), written with an arrow .
The size of $v$ , written $∣ v ∣$ or $v$ , is... ::: its length with direction discarded; for velocity this is called the speed.
Adding two arrows means... ::: laying them tip-to-tail; the arrow from start to finish is the sum.
Subtracting $b$ means... ::: adding $- b$ , the same-length arrow pointing the opposite way; and $b + (- b) = 0$ .
The zero vector $0$ is... ::: an arrow of no length (a dot) — what an equal-and-opposite pair adds up to.
Mass $m$ measures... ::: the amount of stuff — how hard an object is to speed up or stop.
Momentum of one object is... ::: $p = m v$ , an arrow along the velocity, longer for more mass.
Total momentum $P$ is... ::: $\sum_{i} m_{i} v_{i}$ , all the little momentum arrows added tip-to-tail.
An internal force is... ::: a push between two members of the chosen system, written $f_{ij}$ .
An external force is... ::: a push on a system member from outside the system, $F^{e x t}$ .
Net external force $F_{e x t}^{n e t}$ is... ::: $\sum_{i} F_{i}^{e x t}$ , every external push on every member added as arrows.
Why internal forces cancel... ::: they come in twins $f_{j i} = - f_{ij}$ , so $\sum_{i} \sum_{j \neq = i} f_{ij} = 0$ .
$\frac{d p}{d t}$ means... ::: how fast momentum is changing right now (rate of change per instant).
Newton's 2nd law in momentum form is... ::: $F = \frac{d p}{d t}$ — a force is whatever changes momentum over time.

Ready? Then head to the parent topic and the master equation will read like plain English.