Mathematics/Lessons/Modelling in Mechanics/Quantities & SI Units
Mechanics · Modelling in Mechanics · Year 1

Quantities & SI Units

Every mechanics problem is built from a small set of measurable quantities. Knowing which are vectors, which are scalars, and which units they carry is the groundwork everything else in the course rests on.

8 min read 3 worked examples

Scalar and vector quantities

Mechanics describes how objects move and why, and every quantity you use to do that falls into one of two families. A scalar quantity has a size (magnitude) but no direction — mass, time, speed, distance and energy are all scalars. A vector quantity has both a magnitude and a direction — displacement, velocity, acceleration, force and momentum are all vectors. Confusing the two is one of the most common sources of error in the whole course, because everyday language often blurs a distinction mechanics insists on.

Distance vs displacement, speed vs velocity

Distance is the total length of path travelled — a scalar, always positive, always increasing while something moves. Displacement is the straight-line change in position from start to finish, together with a direction — a vector, which can be negative or even zero if you end up back where you started. The same distinction separates speed (scalar, how fast) from velocity (vector, how fast and which way). A runner who completes one lap of a 400 m track has travelled a distance of 400 m but has a displacement of 0 m.

Vectors are written in a few ways in this course: as a bold or underlined letter ($\mathbf{v}$ or $\underline{v}$), as a magnitude with a stated direction ("$5\ \text{m s}^{-1}$ due north"), or — once you meet them properly — in component form using $\mathbf{i}$ and $\mathbf{j}$. For now, the key habit is to ask of every quantity: does direction matter here? If yes, it is a vector, and a sign or a stated direction is part of a complete answer, not an optional extra.

The SI base units used in mechanics

Every quantity in mechanics is built from three base SI units:

kilogram (kg) — the unit of mass
metre (m) — the unit of length or distance
second (s) — the unit of time

Everything else — velocity, acceleration, force, momentum, energy — is a derived unit, built by combining these three according to how the quantity is defined.

Velocity / speed: $\text{m s}^{-1}$ — displacement per unit time
Acceleration: $\text{m s}^{-2}$ — velocity per unit time
Force / weight: newtons, $\text{N}$, where $1\ \text{N} = 1\ \text{kg m s}^{-2}$
Momentum: $\text{kg m s}^{-1}$ — mass times velocity
Moment: newton-metres, $\text{N m}$ — force times perpendicular distance, met properly in a later topic
Where the newton comes from

Newton's second law, met properly in a later lesson, states $F = ma$. A force of one newton is defined as exactly the force needed to give a mass of $1\ \text{kg}$ an acceleration of $1\ \text{m s}^{-2}$. That is why $\text{N} = \text{kg m s}^{-2}$ — the unit is not arbitrary, it falls straight out of the defining equation.

Deriving units from a formula

If you are ever unsure of a quantity's units, substitute the SI units of the quantities on the right-hand side of its defining formula and simplify. This works for any formula, however unfamiliar.

MethodFind the units of momentum and of kinetic energy
1
Momentum is defined as $p = mv$.
Mass times velocity.
2
Substitute units: $[\text{kg}] \times [\text{m s}^{-1}] = \text{kg m s}^{-1}$.
Multiply the units directly.
3
Kinetic energy is defined as $E_k = \tfrac12 mv^2$.
Numbers like $\tfrac12$ have no units.
4
Substitute units: $[\text{kg}] \times [\text{m s}^{-1}]^2 = \text{kg m}^2\text{s}^{-2}$.
Square the velocity unit too.
▪ derived

Converting units

Exam questions mix units freely — speeds in $\text{km h}^{-1}$, masses in grams, distances in centimetres — and every formula in this course expects SI units. Converting first, before substituting into any formula, avoids errors that are otherwise very easy to make.

Worked example 1

Converting km h⁻¹ to m s⁻¹
A car travels at $108\ \text{km h}^{-1}$. Express this speed in $\text{m s}^{-1}$.
MethodConvert kilometres to metres and hours to seconds
1
$108\ \text{km h}^{-1}$ means $108$ kilometres travelled every hour.
Restate what the unit means.
2
Convert kilometres to metres: $108 \times 1000 = 108\,000\ \text{m}$ per hour.
$1\ \text{km} = 1000\ \text{m}$.
3
Convert hours to seconds: $1$ hour $= 3600\ \text{s}$.
$60 \times 60$.
4
Divide: $108\,000 \div 3600 = 30\ \text{m s}^{-1}$.
Metres travelled per second.
▪ solved
The shortcut

Dividing by $3.6$ converts $\text{km h}^{-1}$ directly to $\text{m s}^{-1}$, since $1000 \div 3600 = \tfrac{1}{3.6}$. Here $108 \div 3.6 = 30\ \text{m s}^{-1}$, matching the longer working above. Multiply by $3.6$ to go the other way.

Worked example 2

Classifying scalars and vectors
State whether each of the following is a scalar or a vector quantity: (a) the mass of a car, (b) the displacement of a cyclist from home, (c) the speed of a train, (d) the tension in a rope.
MethodAsk whether direction is part of the quantity
1
(a) Mass has a size only, no direction: scalar.
Kilograms carry no direction.
2
(b) Displacement is defined as a distance in a stated direction from a fixed point: vector.
Direction is built into the definition.
3
(c) Speed is the magnitude of velocity, with no direction attached: scalar.
Compare with velocity, its vector counterpart.
4
(d) Tension is a force, and every force acts in a particular direction along the string: vector.
All forces are vectors.
▪ solved

Worked example 3

A harder unit derivationChallenge
The power delivered by an engine is defined as $P = Fv$, where $F$ is a driving force and $v$ is speed. Show that the SI unit of power, the watt, is equivalent to $\text{kg m}^2\text{s}^{-3}$.
MethodSubstitute units into P = Fv, then expand the units of force itself
1
The unit of force is the newton, and $1\ \text{N} = 1\ \text{kg m s}^{-2}$.
Force is not a base unit, so expand it too.
2
Substitute into $P = Fv$: the units of $P$ are $[\text{kg m s}^{-2}] \times [\text{m s}^{-1}]$.
Multiply force's unit by speed's unit.
3
Multiply the metre powers and second powers separately: $\text{m}^{1+1} = \text{m}^2$ and $\text{s}^{-2-1} = \text{s}^{-3}$.
Add the indices.
4
So the unit of power is $\text{kg m}^2\text{s}^{-3}$, which is exactly the watt, $\text{W}$.
As required.
▪ shown
Why this matters beyond mechanics

This kind of derivation — reducing a named unit like the watt or the joule back to $\text{kg}$, $\text{m}$ and $\text{s}$ — is called dimensional analysis. It is a genuine check you can run on any formula you are unsure of: if the units on both sides do not match, the formula is wrong, however plausible it looks.

Common mistake

Substituting values into a formula before converting to SI units is the single most common error here — a speed left in $\text{km h}^{-1}$ or a mass left in grams will silently make every later answer wrong by a factor of $1000$ or $3.6$. Convert everything to kilograms, metres and seconds as the very first step, before any other working.

Check yourself

(a) Convert $54\ \text{km h}^{-1}$ to $\text{m s}^{-1}$. (b) State whether acceleration is a scalar or a vector, and explain why in one sentence.

(a) $54 \div 3.6 = 15\ \text{m s}^{-1}$.

(b) Acceleration is a vector: it is the rate of change of velocity, and velocity itself has a direction, so a change in velocity — speeding up, slowing down or turning — is described by both a size and a direction.

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