Visual walkthrough — SI units — seven base units and all derived units

This page is the flagship visual walkthrough for the parent topic. It leans on ideas from Dimensional Analysis, and the physics equations come from Newton's Laws of Motion, Kinematics — velocity and acceleration and Work, Energy and Power.

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Step 1 — The three starter bricks and what "$[\cdot ]$" means

WHAT. Before any recipe, we lay out our three mechanics bricks and give ourselves one piece of shorthand.

The three bricks we need for all of mechanics:

Brick	What it measures	Unit symbol
Length	how far	$\text{m}$ (metre)
Mass	how much stuff / inertia	$\text{kg}$ (kilogram)
Time	how long	$\text{s}$ (second)

WHY. Everything below is just these three coloured bars getting stacked and stretched. If you can hold "metre, kilogram, second" in your head as three physical bars, the algebra later is just bookkeeping of those bars.

PICTURE. Three separate bars — magenta = metre, violet = kilogram, orange = second. They do not overlap: no bar can be built from the others. That independence is exactly why they are base.

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Step 2 — Snapping bricks together: what a power of a unit looks like

WHAT. Our recipes will use things like ${\text{s}}^{-2}$ and ${\text{m}}^2$. Let us see what these mean before they appear in a formula.

WHY. The single most confusing thing for beginners is the minus sign in ${\text{s}}^{-2}$. It is not mysterious: ${\text{s}}^{-2}$ is just $1/{\text{s}}^2$, i.e. a division written as a negative exponent so we can multiply everything in one line instead of stacking fractions.

PICTURE. Left: two orange bars forming an ${\text{s}}^2$ square. Right: the same square with a red "÷" slash and the label ${\text{s}}^{-2}$ — same object, flipped from "multiply" to "divide".

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Step 3 — Velocity: the first snap (m ÷ s)

WHAT. Our first derived unit. Velocity answers "how far per how long?" — so we divide a length brick by a time brick.

$$v=\underset{\text{distance per time}}{\underbrace{\frac{d}{t}}}\Rightarrow [v]=\frac{[d]}{[t]}=\frac{\text{m}}{\text{s}}={\text{m⋅s}}^{-1}$$

Term by term:

• $d$ = distance travelled → its unit is $\text{m}$ (the magenta brick).
• $t$ = time taken → its unit is $\text{s}$ (the orange brick).
• The bar   ​ = "per" → it turns $\text{s}$ into ${\text{s}}^{-1}$.

WHY this equation and not another? Because velocity is defined as distance over time — this comes straight from Kinematics — velocity and acceleration. We are not choosing a unit for velocity; the definition hands it to us. That is the whole game: the physics equation IS the recipe.

PICTURE. A magenta metre-bar sitting on top of an orange second-bar, a division bar between them, producing the label ${\text{m⋅s}}^{-1}$.

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Step 4 — Acceleration: dividing by time again

WHAT. Acceleration answers "how fast is the velocity itself changing, per second?" So we take velocity's unit and divide by time once more.

$$a=\frac{\Delta v}{t}\Rightarrow [a]=\frac{[v]}{[t]}=\frac{{\text{m⋅s}}^{-1}}{\text{s}}={\text{m⋅s}}^{-2}$$

Term by term:

• $\Delta v$ = change in velocity → unit ${\text{m⋅s}}^{-1}$ (from Step 3).
• Dividing by $t$ again → adds one more ${\text{s}}^{-1}$, giving ${\text{s}}^{-2}$.

WHY the double "per second"? Read ${\text{m⋅s}}^{-2}$ out loud as "metres per second, per second". The first "per second" is the speed; the second "per second" is how fast that speed grows. Two time-divisions stacked — that is what the exponent $-2$ is counting.

PICTURE. Step 3's ${\text{m⋅s}}^{-1}$ label fed through a second orange time-divider, landing on ${\text{m⋅s}}^{-2}$. A little car speeds up along the bottom to remind you what accelerating looks like.

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Step 5 — The newton: force = mass × acceleration

WHAT. Now the star of mechanics. Newton's second law says force is mass times acceleration — so we multiply the mass brick by the acceleration unit we just built.

$$F=ma\Rightarrow [F]=[m][a]=\text{kg}\cdot {\text{m⋅s}}^{-2}=\underset{\text{define this as }1\text{N}}{\underbrace{{\text{kg⋅m⋅s}}^{-2}}}$$

Term by term:

• $m$ = mass → $\text{kg}$ (the violet brick).
• $a$ = acceleration → ${\text{m⋅s}}^{-2}$ (built in Step 4).
• The dot "$\cdot$" = "times" → we stack the violet brick onto the ${\text{m⋅s}}^{-2}$ block.
• The result is named $1\text{N}$ (newton) purely for convenience — it is still just the three bricks.

WHY it is derived, not base. There is no separate "force standard" locked in a vault. Force's unit falls out of $F=ma$ (see Newton's Laws of Motion). That is why the parent note insists the newton is a derived unit — mass, length, time do all the work.

PICTURE. The violet $\text{kg}$ brick clicking onto the orange-and-magenta ${\text{m⋅s}}^{-2}$ block like a LEGO piece, with a big banner "$=1\text{N}$".

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Step 6 — The joule: force acting through a distance

WHAT. Energy (work) is force applied along a distance. So multiply the newton by one more metre brick.

$$W=Fd\Rightarrow [W]=[F][d]=\underset{\text{newton}}{\underbrace{{\text{kg⋅m⋅s}}^{-2}}}\cdot \text{m}={\text{kg⋅m}}^2{\text{⋅s}}^{-2}\equiv 1\text{J}$$

Term by term:

• $F$ = force → ${\text{kg⋅m⋅s}}^{-2}$ (Step 5).
• $d$ = distance pushed → one more $\text{m}$, so the $\text{m}$ becomes ${\text{m}}^2$.
• Notice ${\text{s}}^{-2}$ did not change — we multiplied by a length, not a time.

WHY only the metre-count grows. Multiplying by a length brick only touches the length pile. The mass pile and time pile are untouched — a clean, visual reason $\text{m}$ becomes ${\text{m}}^2$ while $\text{kg}$ and ${\text{s}}^{-2}$ stay put. This is the energy unit from Work, Energy and Power.

PICTURE. The newton block with a second magenta metre-brick snapping onto its single metre, merging into an ${\text{m}}^2$ square, banner "$=1\text{J}$".

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Step 7 — The watt: energy per second

WHAT. Power is how fast energy is delivered — energy divided by time. Divide the joule by one time brick.

$$P=\frac{W}{t}\Rightarrow [P]=\frac{[W]}{[t]}=\frac{{\text{kg⋅m}}^2{\text{⋅s}}^{-2}}{\text{s}}={\text{kg⋅m}}^2{\text{⋅s}}^{-3}\equiv 1\text{W}$$

Term by term:

• $W$ = energy → ${\text{kg⋅m}}^2{\text{⋅s}}^{-2}$ (Step 6).
• Dividing by $t$ → adds one more ${\text{s}}^{-1}$: the exponent drops from $-2$ to $-3$.

WHY the time exponent alone moves. Same rule as Step 4 in reverse: a time-division only feeds the time pile. So ${\text{s}}^{-2}\to {\text{s}}^{-3}$ while $\text{kg}$ and ${\text{m}}^2$ sit still.

PICTURE. The joule block passing through a single orange time-divider, its time-exponent counter ticking from $-2$ to $-3$, banner "$=1\text{W}$".

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Step 8 — The degenerate & edge cases (dimensionless and "per something")

WHAT. Not every quantity grows a new brick. Two special cases every reader will meet:

1. Dimensionless quantities — pure numbers with no bricks at all.
2. "Per something" units — like frequency, where the length and mass piles are empty.

$$\text{(a)}\left[\frac{1}{2}\right]=1(\text{no units at all})\text{(b)}[\text{frequency}]=\frac{1}{\text{s}}={\text{s}}^{-1}\equiv 1\text{Hz}$$

• $\frac{1}{2}$, $\pi$, angles in radians, refractive index: all bricks cancel → dimensionless. That is why the parent note's Example 2 could ignore the $\frac{1}{2}$ in $\frac{1}{2}m{v}^2$.
• Frequency $=1/\text{period}$: the mass pile and length pile are empty; only a single time-division remains. A perfectly valid unit built from zero metre and zero kilogram bricks.

WHY show these. A recipe machine must handle the empty recipe too. If you ever get all bricks cancelling, that is not an error — it means the quantity is a pure number. And a unit like $\text{Hz}={\text{s}}^{-1}$ proves a "derived" unit can use just one of the seven bricks.

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The one-picture summary

One flow, three starting bricks, four snaps:

$$\underset{\text{base bricks}}{\underbrace{\text{m},\text{kg},\text{s}}}\underset{÷t}{\overset{\text{÷ time}}{\to }}{\text{m⋅s}}^{-1}\underset{÷t}{\overset{\text{÷ time}}{\to }}{\text{m⋅s}}^{-2}\underset{\times m}{\overset{\text{× mass}}{\to }}\text{N}\underset{\times d}{\overset{\text{× length}}{\to }}\text{J}\underset{÷t}{\overset{\text{÷ time}}{\to }}\text{W}$$

Recall Feynman retelling in plain words

I started with three toy bricks: a ruler (metre), a weight (kilogram), and a stopwatch (second). First I asked "how fast?" and divided ruler by stopwatch → that is speed, $\text{m/s}$. Then I asked "how fast is the speed changing?" and divided by the stopwatch again → acceleration, ${\text{m/s}}^2$. Then Newton whispered "push = weight × acceleration", so I clicked the weight brick on and got the newton. Then I pushed something across the room — force times distance — so I clicked one more ruler on, and the single metre became a metre-squared: that is the joule, energy. Then I asked "how fast is energy being spent?" — divided by the stopwatch once more — and got the watt, power. Along the way I noticed two odd fellows: pure numbers like $\frac{1}{2}$ that carry no brick, and frequency that carries only the stopwatch. The punchline: I never invented a new brick even once. Every fancy unit was just the same three toys snapped in a pattern that the physics equation told me to use.

Recall

Which base bricks make the newton? ::: ${\text{kg⋅m⋅s}}^{-2}$ — mass × length ÷ time². What operation turns the joule into the watt? ::: Dividing by time (× ${\text{s}}^{-1}$), so ${\text{s}}^{-2}\to {\text{s}}^{-3}$. Why does force count as derived, not base? ::: It falls out of $F=ma$; no separate standard exists. What is the unit of a dimensionless quantity? ::: None — all bricks cancel (a pure number). Frequency uses how many of the seven bricks? ::: One — just the second, ${\text{s}}^{-1}$.