Foundations — Newton's first law — inertia, operational definition of force

This page assumes nothing. Every letter, arrow, and squiggle the parent note used is built here from the ground up, in the order that lets each one stand on the last.

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1. Position — "where a thing is"

Before motion, we need location. Pick a starting dot (call it the origin) and lay down two number-lines crossing at it: one going right (call it $x$), one going up (call it $y$). Any point is then "so-many steps right, so-many steps up."

Look at Figure s01: the black corner is the origin, the two black arrows are the $x$ and $y$ axes, and the red dot is the body. The dashed black lines are "measuring tapes" dropping the dot down to $x=3.5$ and across to $y=2.4$ — that pair of numbers is its position.

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2. The arrow — what a vector is

Many quantities in physics need two facts at once: how much and which way. "5 metres north" is different from "5 metres east." We draw such a quantity as an arrow: its length is the "how much," the way it points is the "which way."

The little arrow-hat $\stackrel{⃗}{}$ is the parent note's way of quietly saying "this thing has a direction — don't forget it." Every time you see $\vec{F}$ or $\vec{v}$, read it as "arrow."

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3. Adding and subtracting arrows

Before we can talk about change, we must know how to combine arrows. This is the one piece of arrow-arithmetic the whole topic needs.

Look at Figure s02: to add two arrows, slide the second so its tail sits on the first's tip (tip-to-tail); the arrow from the very start to the very end is the sum. To subtract ${\vec{v}}_i$ from ${\vec{v}}_f$ (written ${\vec{v}}_f-{\vec{v}}_i$), you reverse ${\vec{v}}_i$ (flip it $180°$) and add that — the red arrow from the tip of ${\vec{v}}_i$ to the tip of ${\vec{v}}_f$ is the difference.

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4. Time — the coordinate that lets things change

Position alone is a frozen snapshot. Motion needs a second thing to plot against: time.

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5. Velocity — the arrow the whole law is about

Now combine position + time + arrow. Velocity is the arrow that says how the position-dot is changing as time ticks: how fast, and in which direction.

Look at Figure s03: the black dot is the body now; the red arrow $\vec{v}$ points the way it is heading, and its length is how fast. The faint dot ahead shows where it will be after a little time $t$.

The units: velocity is measured in metres per second, written $\text{m/s}$ — "how many metres of position gained each second of time $t$."

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6. "Change" and the symbol $\Delta$

The law is about change in motion. We need a symbol for "how much something changed." That symbol is the Greek capital delta, $\Delta$.

The tiny time-interval $\Delta t$ — final clock-reading minus start clock-reading — is how we measure how long a change took. In a quick-yank trick (like whipping a cloth from under dishes), $\Delta t$ is tiny, so even a real friction force acts too briefly to change the velocity much.

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7. Acceleration — the symptom of a force

If velocity is the motion-arrow, acceleration is "how fast that arrow is changing." It is itself an arrow.

There are three ways an arrow can change, so three ways to accelerate:

What changes	Picture	Example
Arrow gets longer	speeding up	car accelerating
Arrow gets shorter	slowing down	braking bus
Arrow turns	same length, new direction	puck on a string

All three count as acceleration — the third one is the sneaky case the parent note keeps warning about.

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8. Force — the cause, written $\vec{F}$

Now the star of the parent note. A force is a push or pull. You never see it directly; you infer it from acceleration.

Look at Figure s04: four black force-arrows push a body left, right, up, and down. Because thrust cancels drag and lift cancels weight, they add up to the zero vector — the red dot marks ${\vec{F}}_{\text{net}}=\vec{0}$.

The core statement of the whole topic, now fully readable symbol-by-symbol: ${\vec{F}}_{\text{net}}=\vec{0}⟺\vec{v}=\text{constant}$

Read it aloud: "The net push-arrow is the zero vector if and only if the motion-arrow never changes." The double arrow $⟺$ means both directions are true: no net force guarantees constant velocity, and constant velocity guarantees no net force.

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9. Mass — the number that measures stubbornness

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10. Frame of reference — from where we watch

The last hidden idea. Every measurement of position and velocity is made from somewhere — standing on the ground, or riding a bus. That viewpoint is a frame of reference.

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Prerequisite map

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Equipment checklist

Test yourself — cover the right side and answer each before revealing.

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Connections

• Newton's first law — inertia, operational definition of force — the parent this page equips you for.
• Newton's second law — F=ma — turns the force-arrow into a number using mass and acceleration.
• Newton's third law — action-reaction — force-arrows always come in pairs.
• Inertial vs non-inertial frames — where the first law is allowed to live.
• Friction — the hidden force-arrow that fooled Aristotle.
• Uniform circular motion — the turning-arrow case in full.
• Pseudo-forces — invented arrows in non-inertial frames.
• Mass vs weight — the stubbornness number vs the gravity arrow.