Quantities and Units

What you'll learn

Recognise the SI units used in AS Mechanics.
Distinguish between related quantities such as distance and displacement.
Convert common units into metres, seconds and kilograms before calculating.
Use units to check constant-acceleration answers.

Why units matter in Mechanics

In Mechanics, the numbers you calculate are only meaningful if you know what they measure. Saying a car travels “20” is incomplete: 20 m, 20 km, 20 m s⁻¹ and 20 seconds are completely different things.

Definition

Quantity, unit and SI unit

A quantity is something measurable, such as mass, time, velocity or force.
A unit tells you the scale used to measure the quantity, such as metres or seconds.
An SI unit is a standard unit from the International System of Units, used so that calculations are consistent.

The most important SI units in this topic are metres, seconds and kilograms. Units such as m s⁻¹ and m s⁻² are built from these.

A unit relationship diagram showing how velocity, acceleration and force units are built from metres, seconds and kilograms.

Diagram linking displacement, velocity, acceleration and force units

Example

Identifying mechanics quantities and units

A short mechanics description says: “A trolley of mass 4 kg moves 12 m in 3 seconds. Its acceleration is 2 m s⁻².” Identify the quantity measured by each unit.

The unit kg measures mass, so 4 kg is the trolley’s mass.
The unit m measures distance or displacement, so 12 m is a length-type quantity.
The unit seconds measures time, so 3 seconds is the time taken.
The unit m s⁻² measures acceleration, because acceleration means change in velocity per second.
If a force were mentioned, its SI unit would be newtons, written N.

Distance, displacement, speed and velocity

Some words in Mechanics look similar but mean different things.

Definition

Distance and displacement

Distance is the total length travelled. It is a scalar, so it has size only.
Displacement is the change in position from the starting point. It is a vector, so it has size and direction.

Definition

Speed and velocity

Speed is the rate of change of distance. It is a scalar.
Velocity is the rate of change of displacement. It is a vector, so direction matters.

For motion in a straight line, you usually choose one direction as positive. Motion in the opposite direction is then negative.

A one-dimensional sign convention showing positive and negative directions for displacement, velocity and acceleration.

One-dimensional sign convention for displacement and velocity

Key Idea

Direction matters

In constant-acceleration questions, displacement, velocity and acceleration can be positive or negative depending on the chosen direction.

Example

Distance, displacement, speed and velocity

A cyclist travels 80 m east, then 30 m west. The journey takes 25 seconds. Find the distance, displacement, average speed and average velocity.

A route diagram for the cyclist showing the total path travelled and the unknown resultant displacement.

Distance is the total length travelled, regardless of direction:
$80 + 30 = 110$
The distance travelled is 110 m.
Take east as positive. Displacement is final position relative to the start:
$80 - 30 = 50$
The displacement is 50 m east.
Average speed uses distance divided by time:
$11025=4.4\frac{110}{25} = 4.4$
The average speed is 4.4 m s⁻¹.
Average velocity uses displacement divided by time:
$5025=2\frac{50}{25} = 2$
The average velocity is 2 m s⁻¹ east.

Common Mistake

Using distance when the question asks for displacement

If an object goes forwards and then backwards, do not just add all the lengths unless the question asks for distance or speed. For displacement or velocity, include direction.

Derived units

Some units are made by combining other units. These are called derived units.

Velocity is displacement divided by time, so its unit is metres per second, written m s⁻¹.

Acceleration is change in velocity divided by time, so its unit is metres per second per second, written m s⁻².

Force is measured in newtons, written N.

Definition

The newton

One newton is the force needed to give a mass of 1 kg an acceleration of 1 m s⁻². So 1 N is equivalent to 1 kg m s⁻².

A force diagram illustrating the definition of one newton using a 1 kg mass accelerating at 1 m s⁻².

Example

Finding the unit of acceleration

A particle’s velocity changes from 3 m s⁻¹ to 15 m s⁻¹ in 4 seconds. Find its acceleration and give the correct unit.

Acceleration is change in velocity divided by time:
$\frac{v - u}{t}$
Substitute the values:
$\frac{15 - 3}{4}$
Calculate the acceleration:
$a = 3$
The answer is 3 m s⁻² because acceleration is measured in metres per second squared.

Converting to SI units

Before using a Mechanics formula, convert quantities into SI units unless the question clearly says otherwise.

Useful conversions:

1 km = 1000 m
1 cm = 0.01 m
1 minute = 60 seconds
1 hour = 3600 seconds
To convert km h⁻¹ to m s⁻¹, multiply by $518\frac{5}{18}$ .

A conversion chain showing why kilometres per hour are converted to metres per second by multiplying by 5/18.

Tip

Quick speed conversion

For speeds, 18 km h⁻¹ is the same as 5 m s⁻¹. So 54 km h⁻¹ is 15 m s⁻¹.

Example

Converting before calculating

A runner covers 1.5 km in 5 minutes. Find the average speed in m s⁻¹.

A setup diagram for converting the runner’s distance and time before finding average speed.

Convert the distance into metres:
$1.5 \times 1000 = 1500$
Convert the time into seconds:
$\times 60 = 300$
Use average speed equals distance divided by time:
$1500300=5\frac{1500}{300} = 5$
The average speed is 5 m s⁻¹.

Common Mistake

Mixing kilometres and seconds

Do not calculate with 1.5 km and 300 seconds together. The number you get would not be in m s⁻¹, so it would not match the standard Mechanics units.

Units in constant acceleration formulae

Definition

Constant acceleration

Constant acceleration means the acceleration stays the same throughout the motion. In straight-line motion, this allows you to use the standard constant-acceleration formulae.

The common symbols are:

$u$ : initial velocity, in m s⁻¹
$v$ : final velocity, in m s⁻¹
$a$ : acceleration, in m s⁻²
$t$ : time, in seconds
$s$ : displacement, in metres

Common Mistake

The two meanings of s

The symbol $s$ often means displacement, but the unit symbol s means seconds. Read the context carefully.

One useful formula is:

v = u + a t

This makes sense in units: $a t$ has units m s⁻² multiplied by seconds, giving m s⁻¹, so it can be added to velocity.

Example

Using units in a constant-acceleration calculation

A particle has initial velocity 4 m s⁻¹ and accelerates uniformly at 1.5 m s⁻² for 8 seconds. Find its final velocity.

A constant-acceleration motion diagram showing the given initial velocity, acceleration and time, with final velocity unknown.

Identify the quantities:
$4,\quad a = 1.5,\quad t = 8$
Use the formula for final velocity:
$v = u + a t$
Substitute the values:
$\times 8$
Calculate:
$v = 16$
The final velocity is 16 m s⁻¹.

Checking formulae using units

Unit checking is a powerful way to spot mistakes. You can only add or subtract quantities with the same units.

For example, in the formula

\frac{1}{2}at^2

both terms on the right must have units of metres, because $s$ is displacement.

A unit-checking diagram showing that both terms in the displacement formula have units of metres.

Example

Checking the units in a displacement formula

Show that $\frac{1}{2}at^2$ is consistent with SI units.

The unit of $u$ is m s⁻¹ and the unit of $t$ is seconds.
So $u t$ has units of metres:
$s−1×s=m\text{m s}^{-1} \times \text{s} = \text{m}$
The unit of $a$ is m s⁻². The unit of $t^2$ is s².
So $at^2$ also has units of metres:
$s−2×s2=m\text{m s}^{-2} \times \text{s}^2 = \text{m}$
Since both terms have units of metres, the formula is consistent.

Exam technique

In the exam

Write down units whenever you calculate a physical quantity, especially final answers.
Convert to SI units before using constant-acceleration formulae: metres, seconds, kilograms, m s⁻¹ and m s⁻².
Use direction signs carefully: choose a positive direction and keep it consistent.
Check whether your answer’s unit matches the quantity asked for.

Self review

Check yourself

What is the difference between distance and displacement?
Why is acceleration measured in m s⁻² rather than m s⁻¹?
If a speed is given in km h⁻¹, what should you do before using it in a Mechanics formula?

Quantities and Units

What you'll learn

Recognise the SI units used in AS Mechanics.
Distinguish between related quantities such as distance and displacement.
Convert common units into metres, seconds and kilograms before calculating.
Use units to check constant-acceleration answers.

Why units matter in Mechanics

Definition

Quantity, unit and SI unit

A quantity is something measurable, such as mass, time, velocity or force.
A unit tells you the scale used to measure the quantity, such as metres or seconds.
An SI unit is a standard unit from the International System of Units, used so that calculations are consistent.

The most important SI units in this topic are metres, seconds and kilograms. Units such as m s⁻¹ and m s⁻² are built from these.

A unit relationship diagram showing how velocity, acceleration and force units are built from metres, seconds and kilograms.

Diagram linking displacement, velocity, acceleration and force units

Example

Identifying mechanics quantities and units

A short mechanics description says: “A trolley of mass 4 kg moves 12 m in 3 seconds. Its acceleration is 2 m s⁻².” Identify the quantity measured by each unit.

The unit kg measures mass, so 4 kg is the trolley’s mass.
The unit m measures distance or displacement, so 12 m is a length-type quantity.
The unit seconds measures time, so 3 seconds is the time taken.
The unit m s⁻² measures acceleration, because acceleration means change in velocity per second.
If a force were mentioned, its SI unit would be newtons, written N.

Distance, displacement, speed and velocity

Some words in Mechanics look similar but mean different things.

Definition

Distance and displacement

Distance is the total length travelled. It is a scalar, so it has size only.
Displacement is the change in position from the starting point. It is a vector, so it has size and direction.

Definition

Speed and velocity

Speed is the rate of change of distance. It is a scalar.
Velocity is the rate of change of displacement. It is a vector, so direction matters.

For motion in a straight line, you usually choose one direction as positive. Motion in the opposite direction is then negative.

A one-dimensional sign convention showing positive and negative directions for displacement, velocity and acceleration.

One-dimensional sign convention for displacement and velocity

Key Idea

Direction matters

In constant-acceleration questions, displacement, velocity and acceleration can be positive or negative depending on the chosen direction.

Example

Distance, displacement, speed and velocity

A cyclist travels 80 m east, then 30 m west. The journey takes 25 seconds. Find the distance, displacement, average speed and average velocity.

A route diagram for the cyclist showing the total path travelled and the unknown resultant displacement.

Distance is the total length travelled, regardless of direction:
$80 + 30 = 110$
The distance travelled is 110 m.
Take east as positive. Displacement is final position relative to the start:
$80 - 30 = 50$
The displacement is 50 m east.
Average speed uses distance divided by time:
$11025=4.4\frac{110}{25} = 4.4$
The average speed is 4.4 m s⁻¹.
Average velocity uses displacement divided by time:
$5025=2\frac{50}{25} = 2$
The average velocity is 2 m s⁻¹ east.

Common Mistake

Using distance when the question asks for displacement

If an object goes forwards and then backwards, do not just add all the lengths unless the question asks for distance or speed. For displacement or velocity, include direction.

Derived units

Some units are made by combining other units. These are called derived units.

Velocity is displacement divided by time, so its unit is metres per second, written m s⁻¹.

Acceleration is change in velocity divided by time, so its unit is metres per second per second, written m s⁻².

Force is measured in newtons, written N.

Definition

The newton

One newton is the force needed to give a mass of 1 kg an acceleration of 1 m s⁻². So 1 N is equivalent to 1 kg m s⁻².

A force diagram illustrating the definition of one newton using a 1 kg mass accelerating at 1 m s⁻².

Example

Finding the unit of acceleration

A particle’s velocity changes from 3 m s⁻¹ to 15 m s⁻¹ in 4 seconds. Find its acceleration and give the correct unit.

Acceleration is change in velocity divided by time:
$\frac{v - u}{t}$
Substitute the values:
$\frac{15 - 3}{4}$
Calculate the acceleration:
$a = 3$
The answer is 3 m s⁻² because acceleration is measured in metres per second squared.

Converting to SI units

Before using a Mechanics formula, convert quantities into SI units unless the question clearly says otherwise.

Useful conversions:

1 km = 1000 m
1 cm = 0.01 m
1 minute = 60 seconds
1 hour = 3600 seconds
To convert km h⁻¹ to m s⁻¹, multiply by $518\frac{5}{18}$ .

A conversion chain showing why kilometres per hour are converted to metres per second by multiplying by 5/18.

Tip

Quick speed conversion

For speeds, 18 km h⁻¹ is the same as 5 m s⁻¹. So 54 km h⁻¹ is 15 m s⁻¹.

Example

Converting before calculating

A runner covers 1.5 km in 5 minutes. Find the average speed in m s⁻¹.

A setup diagram for converting the runner’s distance and time before finding average speed.

Convert the distance into metres:
$1.5 \times 1000 = 1500$
Convert the time into seconds:
$\times 60 = 300$
Use average speed equals distance divided by time:
$1500300=5\frac{1500}{300} = 5$
The average speed is 5 m s⁻¹.

Common Mistake

Mixing kilometres and seconds

Do not calculate with 1.5 km and 300 seconds together. The number you get would not be in m s⁻¹, so it would not match the standard Mechanics units.

Units in constant acceleration formulae

Definition

Constant acceleration

Constant acceleration means the acceleration stays the same throughout the motion. In straight-line motion, this allows you to use the standard constant-acceleration formulae.

The common symbols are:

$u$ : initial velocity, in m s⁻¹
$v$ : final velocity, in m s⁻¹
$a$ : acceleration, in m s⁻²
$t$ : time, in seconds
$s$ : displacement, in metres

Common Mistake

The two meanings of s

The symbol $s$ often means displacement, but the unit symbol s means seconds. Read the context carefully.

One useful formula is:

v = u + a t

This makes sense in units: $a t$ has units m s⁻² multiplied by seconds, giving m s⁻¹, so it can be added to velocity.

Example

Using units in a constant-acceleration calculation

A particle has initial velocity 4 m s⁻¹ and accelerates uniformly at 1.5 m s⁻² for 8 seconds. Find its final velocity.

A constant-acceleration motion diagram showing the given initial velocity, acceleration and time, with final velocity unknown.

Identify the quantities:
$4,\quad a = 1.5,\quad t = 8$
Use the formula for final velocity:
$v = u + a t$
Substitute the values:
$\times 8$
Calculate:
$v = 16$
The final velocity is 16 m s⁻¹.

Checking formulae using units

Unit checking is a powerful way to spot mistakes. You can only add or subtract quantities with the same units.

For example, in the formula

\frac{1}{2}at^2

both terms on the right must have units of metres, because $s$ is displacement.

A unit-checking diagram showing that both terms in the displacement formula have units of metres.

Example

Checking the units in a displacement formula

Show that $\frac{1}{2}at^2$ is consistent with SI units.

The unit of $u$ is m s⁻¹ and the unit of $t$ is seconds.
So $u t$ has units of metres:
$s−1×s=m\text{m s}^{-1} \times \text{s} = \text{m}$
The unit of $a$ is m s⁻². The unit of $t^2$ is s².
So $at^2$ also has units of metres:
$s−2×s2=m\text{m s}^{-2} \times \text{s}^2 = \text{m}$
Since both terms have units of metres, the formula is consistent.

Exam technique

In the exam

Write down units whenever you calculate a physical quantity, especially final answers.
Convert to SI units before using constant-acceleration formulae: metres, seconds, kilograms, m s⁻¹ and m s⁻².
Use direction signs carefully: choose a positive direction and keep it consistent.
Check whether your answer’s unit matches the quantity asked for.

Self review

Check yourself

What is the difference between distance and displacement?
Why is acceleration measured in m s⁻² rather than m s⁻¹?
If a speed is given in km h⁻¹, what should you do before using it in a Mechanics formula?

Quantities and Units

What you'll learn

Why units matter in Mechanics

Distance, displacement, speed and velocity

Derived units

Converting to SI units

Units in constant acceleration formulae

Checking formulae using units

Recap questions

You've reached the end

Quantities and Units

What you'll learn

Why units matter in Mechanics

Distance, displacement, speed and velocity

Derived units

Converting to SI units

Units in constant acceleration formulae

Checking formulae using units

Recap questions

You've reached the end

Quantities and Units

What you'll learn

Why units matter in Mechanics

Distance, displacement, speed and velocity

Derived units

Converting to SI units

Units in constant acceleration formulae

Checking formulae using units

Recap questions

You've reached the end

Quantities and Units Revision Guide

Quantities and Units

What you'll learn

Why units matter in Mechanics

Distance, displacement, speed and velocity

Derived units

Converting to SI units

Units in constant acceleration formulae

Checking formulae using units

Recap questions

You've reached the end

Quantities and Units Revision Guide