Foundations — Conservation of linear momentum — derivation from Newton's third law

This page assumes nothing. If the parent note wrote a symbol, we build it here from the ground up, in an order where each idea stands on the one before it. Read top to bottom.

1. A number that also points — the arrow (vector)

Plain speed "5 metres per second" is just a number — it doesn't say which way. Velocity says "5 m/s to the right". That extra "which way" is exactly what the little arrow on top, $\vec{v}$, promises you.

The picture: an arrow drawn on the floor. Look at the orange arrow — its length is "how fast", its pointing is "which way". A short arrow = slow, a long arrow = fast.

Why the topic needs it: momentum is going to be an arrow too, because the direction a bullet flies matters as much as its speed. If we threw away direction, we could never explain why a gun kicks backwards.

2. Splitting an arrow into $x$ and $y$ (components)

$\hat{x}$ is read "x-hat". The hat means "this is a pure direction, length one, no size baggage." So $6\hat{x}$ means "6 units, pointing right."

The picture: the magenta arrow is the shadow the vector casts on the floor ($v_x$); the violet arrow is the shadow it casts on the wall ($v_y$). Together the two shadows rebuild the whole arrow.

Why the topic needs it: the "conserve each axis independently" trick (parent Example 3) is impossible without components.

3. Mass $m$ — how much stuff

The picture: think of two balls, a marble and a bowling ball. Same push, the marble leaps, the bowling ball barely stirs — that reluctance is mass.

Why the topic needs it: momentum is mass times velocity. Mass is the number that says "how much motion-stuff a given speed is worth."

4. Momentum $\vec{p}=m\vec{v}$ — the "motion-money"

The picture: the same velocity arrow, but for a heavy object it is stretched into a longer momentum arrow. A slow truck can carry more momentum than a fast bicycle because mass stretches the arrow.

Why "linear"? It just means straight-line momentum, to distinguish it from spinning (angular) momentum, which the parent topic doesn't touch. For us, "momentum" and "linear momentum" are the same thing.

Why the topic needs it: this is the very quantity the whole chapter says is conserved. It's the "money" in the closed-room analogy.

5. The Greek $\Sigma$ — "add them all up" (sum)

So $\sum _i{m}_i{\vec{v}}_i$ means: take particle 1's momentum, plus particle 2's, plus particle 3's, … all the way through, and add the arrows.

Why the topic needs it: "total momentum" of a system means the summed arrow of every piece. The parent's $\vec{P}={\sum }_i{\vec{p}}_i$ is literally this command.

6. Force $\vec{F}$ — a push or pull (also an arrow)

The picture: a hand shoving a box — the arrow starts at the box (where it's felt) and points the way of the shove.

Why the topic needs it: the whole derivation is about how forces change momentum, and how the two forces in a push cancel.

7. The rate-of-change idea $\frac{d}{dt}$ — the derivative

This is the one piece of genuinely new machinery, so we go slowly.

The picture: a curve of momentum against time. The derivative is the steepness (slope) of that curve at a point — steep means changing fast, flat means not changing at all.

Why "$d$" and not just a subtraction sign? Because $d/dt$ handles the instant-by-instant changing force, whereas a plain "after minus before" would need the force to be constant. Newton's second law is honestly $\vec{F}=d\vec{p}/dt$; the familiar $\vec{F}=m\vec{a}$ is just the special case when $m$ doesn't change.

Why the topic needs it: the whole proof runs on "the derivative of total momentum is zero, therefore total momentum is constant." Without the derivative idea, Steps 3–5 of the parent are meaningless.

8. Before vs after — the prime mark ${}^{\prime }$

Why the topic needs it: conservation is a statement comparing before and after — $m_1{\vec{v}}_1+m_2{\vec{v}}_2=m_1{\vec{v}}_1^{\prime }+m_2{\vec{v}}_2^{\prime }$ — and the prime is how we tell the two moments apart.

9. "Isolated" and "external" — the fine print

The picture: draw a dotted circle around the objects you care about. Arrows that cross the circle are external; arrows entirely inside are internal.

Why the topic needs it: momentum is conserved only for an isolated system. The internal pushes cancel by Newton's third law; only external pushes can change the total.

The prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.

Connections

• Yeh foundations Hinglish mein
• Newton's Third Law — supplies the cancelling internal-force pairs
• Newton's Second Law (momentum form) — where $\vec{F}=d\vec{p}/dt$ lives
• Impulse–Momentum Theorem — adds up force over time instead of instantaneously
• Centre of Mass Motion — the summed momentum $\vec{P}=M{\vec{v}}_{cm}$
• Elastic vs Inelastic Collisions — uses these symbols to compare energy
• Rocket Equation — momentum bookkeeping when mass itself changes