P1, J1MechanicsFoundation

SI Units and Vectors in Mechanics

Mechanics is quantitative, so every quantity carries a unit and many carry a direction as well. Getting both right is not clerical detail — a units check will catch an algebraic error, and treating a vector as a scalar will produce an answer that is confidently wrong.

25 min Quantities and Units

The big picture

Two habits from this lesson pay off across the whole strand. The first is that units can be treated algebraically: if you derive a formula and the units come out as anything other than the units of the quantity you wanted, the derivation is wrong. That check costs seconds and catches errors nothing else will. The second is the sign convention: choose a positive direction, write it down, and stick to it. Almost every "impossible" negative answer in mechanics is a direction being interpreted rather than a value being wrong.

What you'll be able to do

State the SI base units used in mechanics and identify derived units
Use unit analysis to check whether an equation is dimensionally consistent
Classify mechanical quantities as scalar or vector
Apply a consistent sign convention and interpret negative results

Base and derived units

Mechanics uses three SI base units: the $\textbf{metre}$ (m) for length, the $\textbf{kilogram}$ (kg) for mass, and the $\textbf{second}$ (s) for time. Everything else is built from them.

A $\textbf{derived}$ unit is a combination. Velocity is length per time, so $\mathrm{m\,s^{-1}}$ . Acceleration is velocity per time, so $\mathrm{m\,s^{-2}}$ . Force, from $F = ma$ , is $\mathrm{kg\,m\,s^{-2}}$ — which is given the name $\textbf{newton}$ (N).

Note that mass is a base quantity but $\textbf{weight}$ is a force, measured in newtons. A 5 kg object has a weight of about 49 N, and confusing the two is a persistent source of error.

1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}

The newton, from

F = ma

W = mg

Weight in newtons, from mass in kilograms.

Example 1Express the unit of momentum (

mv

) in base units.

1Mass is kg; velocity is

\mathrm{m\,s^{-1}}

2Multiply:

\mathrm{kg} \times \mathrm{m\,s^{-1}}

Answer

\mathrm{kg\,m\,s^{-1}}

Tip — Always convert to SI units before calculating. A speed in km/h or a mass in grams will silently produce an answer that is wrong by a factor of 1000 or 3.6.

Checking equations with units

Units behave algebraically, so both sides of a correct equation must reduce to the same combination. If they do not, the equation is wrong — no exceptions.

The check is quick and catches real mistakes. Take $v^{2} = u^{2} + 2as$ : the left is $\left(\mathrm{m\,s^{-1}}\right)^{2} = \mathrm{m^{2}\,s^{-2}}$ , and the right-hand term $as$ is $\mathrm{m\,s^{-2}} \times \mathrm{m} = \mathrm{m^{2}\,s^{-2}}$ . They agree.

Note what the check cannot do: dimensionless constants are invisible to it. $s = ut + \tfrac12 at^{2}$ and $s = ut + 2at^{2}$ both pass, so a units check proves an equation is not obviously wrong rather than proving it right.

Example 1A student writes

s = \dfrac{v^{2}-u^{2}}{a}

. Check whether it is dimensionally consistent.

1Numerator:

\mathrm{m^{2}\,s^{-2}}

2Denominator:

\mathrm{m\,s^{-2}}

3Dividing gives

\mathrm{m}

, which matches displacement.

AnswerDimensionally consistent — though the correct formula has a factor of 2 in the denominator, which a units check cannot detect.

Why this works

That last point is worth remembering: units verify the $\textit{structure}$ of a formula but never its numerical coefficients. A passing check narrows the possibilities without confirming the answer.

Scalars and vectors in mechanics

Scalars have size only: mass, time, distance, speed, energy, work, power. Vectors have size and direction: displacement, velocity, acceleration, force, weight, momentum.

The pairs that get confused are distance/displacement and speed/velocity. An object returning to its start has travelled a real distance but has zero displacement, and therefore zero average velocity despite a non-zero average speed.

Mechanics questions exploit this. "Find the distance travelled" and "find the displacement" have different answers whenever the direction of motion reverses.

Example 1A particle moves 8 m east then 3 m west. Find the distance travelled and the displacement.

1Distance is the total path length:

8 + 3 = 11

2Displacement is the net change in position:

8 - 3 = 5

m east.

AnswerDistance 11 m; displacement 5 m east.

Tip — Displacement, velocity and acceleration always need a direction stated — as a sign, a compass bearing, or in component form. A bare number is an incomplete answer for a vector.

Sign conventions

In one-dimensional problems, direction is carried by a sign. Choose which direction is positive at the start, state it, and apply it consistently to every quantity in the problem.

The most common choice for vertical motion is upwards positive, which makes $a = -9.8\ \mathrm{m\,s^{-2}}$ and an upward launch velocity positive. Taking downwards positive is equally valid; what is not valid is mixing the two.

A negative answer is information, not an error. A negative displacement means the object finished behind its starting point; a negative velocity means it is moving in the negative direction; a negative acceleration may mean slowing down or speeding up in the negative direction, depending on the velocity’s sign.

v \text{ and } a \text{ same sign} \Rightarrow \text{speeding up}

Opposite signs mean slowing down.

Example 1A particle has velocity

-6\ \mathrm{m\,s^{-1}}

and acceleration

-2\ \mathrm{m\,s^{-2}}

. Describe its motion.

1The velocity is negative, so it moves in the negative direction.

2The acceleration has the same sign as the velocity.

3Same signs mean the speed is increasing.

AnswerMoving in the negative direction and speeding up.

Why this works

Note that "negative acceleration" does not mean "decelerating". It means the acceleration points in the negative direction — whether that speeds the object up or slows it down depends entirely on which way it is already moving.

Think like an examiner

Convert all quantities to SI units before substituting into any formula.
State your sign convention explicitly in one-dimensional problems.
Give vector answers with a direction, not just a magnitude.
Include units on every final answer — a bare number rarely scores.

Common misconceptions

Using mass in grams or speed in km/h without converting.

Convert to kg and

\mathrm{m\,s^{-1}}

first, or every subsequent number is wrong.

Treating weight and mass as the same quantity.

Mass is in kilograms; weight is a force in newtons, equal to

mg

Reading a negative acceleration as always meaning deceleration.

It means acceleration in the negative direction. Compare its sign with the velocity’s.

Concluding a units check proves a formula correct.

It cannot detect a wrong dimensionless constant. It only rules formulas out.

Units and conventions

\mathrm{m},\ \mathrm{kg},\ \mathrm{s}

Base units in mechanics.

1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}

Derived unit of force.

W = mg

Weight from mass.

\text{vector: magnitude + direction}

Needs a sign or bearing.

Stretch yourself

A student proposes that the maximum height reached by a projectile launched vertically at speed $u$ is $h = \dfrac{u^{2}}{g}$ . Use a units check to test this, then determine whether it is correct.

Hint — Check the units first, then derive the true expression using a suvat equation.

Reveal solution

SolutionUnits:

u^{2}

has units

\mathrm{m^{2}\,s^{-2}}

and

g

has

\mathrm{m\,s^{-2}}

, so

\dfrac{u^{2}}{g}

has units

\dfrac{\mathrm{m^{2}\,s^{-2}}}{\mathrm{m\,s^{-2}}} = \mathrm{m}

— a length, so the formula is dimensionally consistent. But that does not make it right. Deriving properly with

v^{2} = u^{2} + 2as

, taking upwards positive so

a = -g

and

v = 0

at the top:

0 = u^{2} - 2gh

, giving

h = \dfrac{u^{2}}{2g}

. The student’s formula is out by a factor of 2. This is precisely the limitation noted earlier: a units check verifies structure but is blind to dimensionless constants, so it can only ever rule a formula out, never confirm it.

\blacksquare

Questions students ask

Why must I convert to SI units?

Because the formulas are derived in SI units. Substituting grams or km/h gives an answer wrong by a factor of 1000 or 3.6, with no warning that anything has gone wrong.

Is a negative acceleration always deceleration?

No. It means the acceleration acts in the negative direction. If the velocity is also negative the object is speeding up; deceleration occurs when velocity and acceleration have opposite signs.

Can a units check prove a formula is right?

No, only that it is not obviously wrong. Dimensionless factors such as

\tfrac12

2\pi

do not affect units, so a wrong coefficient passes the check.

Does it matter which direction I call positive?

Not at all, provided you state it and apply it consistently. Upwards positive and downwards positive give the same physical answer with the signs interpreted accordingly.

Key takeaways

Mechanics uses metres, kilograms and seconds as base units; the newton is derived.
Mass is a scalar in kilograms; weight is a force in newtons equal to $mg$ .
Units can check an equation’s structure but not its dimensionless constants.
Displacement, velocity, acceleration, force and momentum are vectors and need a direction.
Choose a positive direction, state it, and read negative answers as directions rather than errors.