Foundations — Linear momentum p = mv

This page assumes you know nothing. Before you can trust a single line of the parent note Linear momentum $p=mv$, you need to see every symbol it uses. Let's build them one at a time, each on top of the last.

0. The picture we will keep coming back to

Everything in this topic lives on a flat sheet — think of a hockey puck sliding across ice, seen from above. We draw two directions: rightward we call $x$, upward we call $y$. Every quantity we meet will be a number, or an arrow, painted on this sheet.

1. A number vs. an arrow — the very first split

Before any physics, one idea decides how you read every symbol below: is this thing just a size, or a size-with-a-direction?

Why the topic needs this split: momentum is a vector. If you treat it as a plain number you will get collisions wrong, because two things moving in opposite directions carry opposite momentum and must cancel. The whole "signs and components" machinery below exists only because momentum is an arrow, not a bare number.

2. The little arrow: notation $\vec{v}$ and its size $|\vec{v}|$

Read it out loud:

• $\vec{v}$ ::: "the velocity vector — how fast and which way"
• $|\vec{v}|$ or $v$ ::: "the speed — just the number, no direction"

Picture: $\vec{v}$ is the arrow itself; $|\vec{v}|$ is the length of that arrow measured with a ruler.

Why the topic needs it: the parent note writes $\vec{p}=m\vec{v}$ with hats, and later $|\vec{p}|=m|\vec{v}|$ without them. Those are different statements — one about arrows, one about lengths — and you must be able to tell them apart at a glance.

3. Mass $m$ — the "how much stuff" number

Why the topic needs it: $m$ is one of the two ingredients of momentum. Because it is a positive scalar, multiplying a vector $\vec{v}$ by $m$ stretches the arrow but never flips its direction — a fact we lean on constantly.

4. Velocity $\vec{v}$ — the "how fast and which way" arrow

Why the topic needs it: velocity is the direction-carrier of momentum. Since $m$ can only stretch an arrow (Section 3), $\vec{p}$ points in exactly the same direction as $\vec{v}$. The whole "momentum points along the velocity" claim in the parent note is really a statement about this arrow.

5. Multiplying an arrow by a number: what $m\vec{v}$ does

We now have $m$ (a positive number) and $\vec{v}$ (an arrow). The formula asks us to compute $m\vec{v}$. What does "number times arrow" even mean?

• Because $m>0$ always, momentum $\vec{p}=m\vec{v}$ can never point opposite to the velocity.
• The length of the result is $|\vec{p}|=m|\vec{v}|$ — the mass times the speed.

Why the topic needs it: this is the actual operation hidden inside $\vec{p}=m\vec{v}$. Understanding "scale the arrow" makes the vector formula obvious instead of mysterious.

6. Breaking an arrow into $x$ and $y$ pieces — components

A single arrow on the sheet can be rebuilt from two arrows: one purely rightward, one purely upward. These pieces are its components.

Why the topic needs it: because $m$ just scales each shadow, momentum splits the same way: $p_x=m{v}_x,p_y=m{v}_y.$ This is how the parent note handles 2D motion — it never adds arrows directly, it adds their $x$-pieces and their $y$-pieces separately.

7. Getting the arrow's length back: Pythagoras

Once you have the two shadows $p_x$ and $p_y$, how long is the original arrow?

Why the topic needs it: Worked Example 4 in the parent note computes $|\vec{p}|=\sqrt{0.6^2+0.8^2}=1.0$. That single line is this triangle. Without Pythagoras you cannot turn components back into a magnitude.

8. The signs $+$ and $-$ in 1D — direction squeezed onto a line

If everything moves along one line, we do not need arrows at all: we pick one direction as positive and the opposite as negative.

Why the topic needs it: in Worked Example 3, cart B has velocity $-2\text{ m/s}$, giving momentum $-6\text{ kg⋅m/s}$; adding it to $+8$ gives $+2$. Those signs are doing the direction-cancelling. Miss them and every collision total is wrong.

9. Rate of change — reading $\frac{d\vec{p}}{dt}$

The parent note's deepest formula is $\vec{F}=\frac{d\vec{p}}{dt}$. You do not need full calculus to read it.

Why the topic needs it: every mention of Newton's Second Law, impulse, and Conservation of Linear Momentum rests on reading this one rate-of-change symbol. It's a placeholder for "how quickly it changes" — nothing scarier.

Prerequisite map

Equipment checklist

Cover the right side and answer out loud. If any line stumps you, reread that section before opening the parent note.

Connections

• Linear momentum $p=mv$ — the parent topic these foundations feed into.
• Newton's Second Law — where $\frac{d\vec{p}}{dt}$ becomes a law of force.
• Impulse–Momentum Theorem — rearranges the rate-of-change symbol into $\vec{F}dt=d\vec{p}$.
• Conservation of Linear Momentum — needs the signs and components built here.
• Kinetic Energy — another scalar-vs-vector contrast with momentum.