Quantities and units in mechanics

Q: What are the three fundamental SI base quantities and their units used in AS Level Mechanics?

Mass measured in kilograms ( ext{kg}), Length (or Distance) measured in metres ( ext{m}), and Time measured in seconds ( ext{s}).

Q: What is the fundamental difference between a scalar quantity and a vector quantity?

A scalar has magnitude (size) only, whereas a vector has both magnitude and direction.

Q: What is the difference between distance and displacement, and between speed and velocity?

Distance and speed are scalars (total path length and rate of movement), while displacement and velocity are vectors (straight-line change in position with direction and rate of change of displacement with direction).

Q: How do mass and weight differ in mechanics?

Mass is a scalar measuring the amount of matter in $\text{kg}$, while weight is a vector force measuring gravitational pull in $\text{N}$ ($W = mg$).

Mathematics 2210 · CCEA AS Level · 8 min read · Updated: 31 Aug 2026

Welcome to Quantities and Units in Mechanics

Welcome to Mechanics! Whether you love physics or find it a little intimidating, this chapter is your essential toolkit. Mechanics is simply the mathematical study of how objects move and interact with forces in the physical world around us.

Just imagine following a recipe that says "add 5 flour and bake for 20." Five what? Grams, cups, kilograms? Twenty minutes or twenty hours? Units give meaning to numbers. In 1999, NASA lost the \$125 million Mars Climate Orbiter spacecraft because one engineering team used imperial units while another used metric units! Getting our quantities and units right from day one will make the rest of your AS Applied Mathematics journey smooth, logical, and score-maximising.

1. Fundamental SI Units and Derived Units

In mechanics, we describe the physical universe using standard units known as the Système International (SI) base units. In AS Level Mechanics, everything starts from just three fundamental quantities.

The Three Base Quantities

• Mass: The amount of matter in an object, measured in kilograms ($\text{kg}$).
• Length (or Distance): The spatial measurement between two points, measured in metres ($\text{m}$).
• Time: The duration of an event, measured in seconds ($\text{s}$).

Did you know? In everyday life, people often quote their mass in stones or pounds, but in mechanics, mass must always be in kilograms ($\text{kg}$). If an exam question gives you a mass in grams ($\text{g}$), remember to divide by $1000$.

Derived Units in Mechanics

When we combine our base units, we get derived units. You do not need to memorize random letters—derived units simply tell the story of the formula used to calculate them!

• Speed and Velocity: Rate of change of position with respect to time.
Formula: $\text{velocity} = \frac{\text{displacement}}{\text{time}}$
SI Unit: metres per second, written as $\text{m s}^{-1}$ (or $\text{m/s}$).

• Acceleration: Rate of change of velocity with respect to time.
Formula: $\text{acceleration} = \frac{\text{change in velocity}}{\text{time}}$
SI Unit: metres per second squared, written as $\text{m s}^{-2}$.

• Force and Weight: A push or pull exerted on a mass.
Formula (Newton's Second Law): $F = ma$ ($\text{Force} = \text{mass} \times \text{acceleration}$)
SI Unit: Newton ($\text{N}$), where $1\text{ N} = 1\text{ kg m s}^{-2}$.

• Weight: The gravitational force acting on an object's mass.
Formula: $W = mg$ (where $g$ is the acceleration due to gravity, taken as $9.8\text{ m s}^{-2}$ in CCEA mechanics).
SI Unit: Newton ($\text{N}$).

Essential Unit Conversions

Examiners love testing your attention to detail by giving values in non-standard units. Don't worry if this seems tricky at first—just follow these golden rules before plugging numbers into formulas:

• Kilometres to Metres: Multiply by $1000$ ($1\text{ km} = 1000\text{ m}$).
• Centimetres to Metres: Divide by $100$ ($1\text{ cm} = 0.01\text{ m}$).
• Grams to Kilograms: Divide by $1000$ ($500\text{ g} = 0.5\text{ kg}$).
• Hours or Minutes to Seconds: Multiply hours by $3600$ ($1\text{ hour} = 60 \times 60 = 3600\text{ s}$), or multiply minutes by $60$.
• Speed Conversion ($\text{km h}^{-1}$ to $\text{m s}^{-1}$):
Since $1\text{ km} = 1000\text{ m}$ and $1\text{ hour} = 3600\text{ s}$:
$1\text{ km h}^{-1} = \frac{1000\text{ m}}{3600\text{ s}} = \frac{1}{3.6}\text{ m s}^{-1}$
Quick Trick: To turn $\text{km h}^{-1}$ into $\text{m s}^{-1}$, divide by $3.6$. To go the other way ($\text{m s}^{-1}$ to $\text{km h}^{-1}$), multiply by $3.6$.

Key Takeaway: Before starting any calculation, always convert your quantities into base SI units: $\text{kg}$, $\text{m}$, and $\text{s}$.

2. Scalars vs Vectors

Imagine a friend calls and tells you: "I am walking at 5 miles per hour!" You know how fast they are moving, but you have no clue where they are heading. If they add: "...due North," you now have complete information!

Definitions

• Scalar quantity: A quantity that has magnitude (size) only. It has no direction attached to it.
• Vector quantity: A quantity that has both magnitude (size) AND direction.

Mechanics Pairs: Scalar vs Vector

Let's look at the classic mechanics quantities side-by-side:

• Distance (Scalar) vs Displacement (Vector):
Distance is the total length of the path travelled (e.g. "I walked a total of $400\text{ m}$ around the running track").
Displacement is the straight-line distance from the starting position to the final position, including direction (e.g. "After one full lap of the track, my displacement is $0\text{ m}$ because I am back where I started").

• Speed (Scalar) vs Velocity (Vector):
Speed is simply how fast an object is moving, such as $20\text{ m s}^{-1}$.
Velocity is the speed in a specified direction, such as $+20\text{ m s}^{-1}$ upwards or $20\text{ m s}^{-1}$ due East.

• Mass (Scalar) vs Weight (Vector):
Mass is the quantity of matter in an object ($\text{kg}$), which stays the same anywhere in the universe.
Weight is a force ($\text{N}$) acting vertically downwards due to gravity ($W = mg$).

• Other Common Quantities:
- Time: Scalar (it flows only forwards, measured in $\text{s}$).
- Acceleration: Vector (it has both a magnitude and a direction of change, measured in $\text{m s}^{-2}$).
- Force: Vector (every push or pull acts in a specific direction, measured in $\text{N}$).

Understanding Signs in One-Dimensional Motion

When motion is restricted to a straight line (horizontal or vertical), we do not need complex 2D angles. Instead, we use positive ($+$) and negative ($-$) signs to represent opposite directions.

• For vertical motion: usually we define upwards as positive ($+$) and downwards as negative ($-$). Since gravity acts downwards, the acceleration due to gravity is often written as $a = -g = -9.8\text{ m s}^{-2}$.
• For horizontal motion: usually we define right as positive ($+$) and left as negative ($-$).

Key Takeaway: Vectors care about direction; scalars do not. In 1D mechanics, a minus sign tells you the object is moving or accelerating in the opposite direction.

3. Working with Vectors in Two Dimensions

In two dimensions, we describe directions using standard coordinate systems. Mechanics uses two common notations for vectors: unit vectors ($\mathbf{i}$ and $\mathbf{j}$) and column vectors.

Vector Notations

• $\mathbf{i}$ is a unit vector of length $1$ pointing in the positive $x$-direction (horizontal / East).
• $\mathbf{j}$ is a unit vector of length $1$ pointing in the positive $y$-direction (vertical / North).

Any 2D vector $\mathbf{r}$ can be written as:
$\mathbf{r} = x\mathbf{i} + y\mathbf{j}$ or as a column vector $\mathbf{r} = \begin{pmatrix} x \\ y \end{pmatrix}$

Finding Magnitude (Pythagoras' Theorem)

The magnitude of a vector represents its size, length, or scalar value (for example, the magnitude of a velocity vector is its speed). We denote the magnitude of $\mathbf{r}$ as $|\mathbf{r}|$ or simply $r$.

Using Pythagoras' theorem:
$|\mathbf{r}| = \sqrt{x^2 + y^2}$

Finding Direction (Trigonometry)

The direction of a vector is usually described by an angle $\theta$ made with a horizontal line, a vertical line, or a compass bearing.

Using basic right-angled trigonometry:
$\tan(\theta) = \frac{|\text{opposite}|}{|\text{adjacent}|}$
Always sketch a quick right-angled triangle to be completely certain which angle you are calculating!

Key Takeaway: Think of $\mathbf{i}$ and $\mathbf{j}$ as directions on a grid (horizontal and vertical). Magnitude is found using Pythagoras, and direction is found using trigonometry.

4. Step-by-Step Worked Examples

Worked Example 1: Unit Conversion

Problem: A car accelerates uniformly from rest to $90\text{ km h}^{-1}$ in $10\text{ seconds}$. Calculate the acceleration in standard SI units.

Step 1: Convert the final velocity from $\text{km h}^{-1}$ to $\text{m s}^{-1}$.
$v = \frac{90 \times 1000\text{ m}}{3600\text{ s}} = \frac{90}{3.6} = 25\text{ m s}^{-1}$

Step 2: State the known quantities in SI units.
Initial velocity $u = 0\text{ m s}^{-1}$
Final velocity $v = 25\text{ m s}^{-1}$
Time $t = 10\text{ s}$

Step 3: Use the acceleration formula.
$a = \frac{v - u}{t} = \frac{25 - 0}{10} = 2.5\text{ m s}^{-2}$

Worked Example 2: Velocity Vector, Magnitude, and Direction

Problem: A particle moves with velocity vector $\mathbf{v} = (6\mathbf{i} - 8\mathbf{j})\text{ m s}^{-1}$, where $\mathbf{i}$ is due East and $\mathbf{j}$ is due North.
(a) Find the speed of the particle.
(b) Find the bearing on which the particle is travelling.

Solution:
Part (a): Speed is the magnitude of the velocity vector.
$\text{Speed} = |\mathbf{v}| = \sqrt{(6)^2 + (-8)^2}$
$|\mathbf{v}| = \sqrt{36 + 64} = \sqrt{100} = 10\text{ m s}^{-1}$

Part (b): The vector goes $6\text{ m s}^{-1}$ East (positive $x$) and $8\text{ m s}^{-1}$ South (negative $y$).
Let $\alpha$ be the angle below the East axis (the horizontal line):
$\tan(\alpha) = \frac{8}{6} = \frac{4}{3} \implies \alpha = \arctan\left(\frac{4}{3}\right) \approx 53.13^\circ$
To find a three-figure bearing (measured clockwise from North):
$\text{Bearing} = 90^\circ + 53.13^\circ = 143.13^\circ \approx 143^\circ$

Worked Example 3: Resultant Force

Problem: Two forces act on a body: $\mathbf{F_1} = (3\mathbf{i} + 5\mathbf{j})\text{ N}$ and $\mathbf{F_2} = (5\mathbf{i} - 11\mathbf{j})\text{ N}$. Find the resultant force $\mathbf{R}$ and its magnitude.

Solution:
Step 1: Add the vectors component by component.
$\mathbf{R} = \mathbf{F_1} + \mathbf{F_2} = (3\mathbf{i} + 5\mathbf{j}) + (5\mathbf{i} - 11\mathbf{j})$
$\mathbf{R} = (3 + 5)\mathbf{i} + (5 - 11)\mathbf{j} = (8\mathbf{i} - 6\mathbf{j})\text{ N}$

Step 2: Calculate the magnitude of $\mathbf{R}$.
$|\mathbf{R}| = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\text{ N}$

5. Common Mistakes to Avoid

• Forgetting to convert grams to kilograms: Mass must always be in $\text{kg}$. Writing $m = 250$ instead of $m = 0.25\text{ kg}$ causes a factor-of-1000 error in your force calculation!
• Confusing Mass and Weight: Mass is measured in $\text{kg}$; Weight is a force measured in $\text{N}$. Remember that $W = mg$.
• Forgetting the square root or squarerooting negative numbers incorrectly: When squaring components like $-8$, remember $(-8)^2 = +64$. Magnitude is always a positive quantity.
• Giving direction without reference: Never just write "angle is $35^\circ$". Always specify: "$35^\circ$ above the positive $x$-axis", "to the horizontal", or as a "three-figure bearing of $035^\circ$".

6. Chapter Summary & Quick Review

• Base SI Units: Length ($\text{m}$), Mass ($\text{kg}$), Time ($\text{s}$).
• Derived Units: Velocity ($\text{m s}^{-1}$), Acceleration ($\text{m s}^{-2}$), Force ($\text{N} = \text{kg m s}^{-2}$).
• Scalars: Have magnitude only (e.g. distance, speed, mass, time).
• Vectors: Have both magnitude and direction (e.g. displacement, velocity, acceleration, force, weight).
• 2D Vectors: $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$, with magnitude $|\mathbf{r}| = \sqrt{x^2 + y^2}$ and direction angle found via $\tan(\theta) = \frac{y}{x}$.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the three fundamental SI base quantities and their units used in AS Level Mechanics?

Mass measured in kilograms ( ext{kg}), Length (or Distance) measured in metres ( ext{m}), and Time measured in seconds ( ext{s}).

What is a derived unit in mechanics, and what are the SI units for velocity, acceleration, and force?

A derived unit is formed by combining base units. Velocity is measured in $\text{m s}^{-1}$, acceleration in $\text{m s}^{-2}$, and force in Newtons ($\text{N}$, where $1\text{ N} = 1\text{ kg m s}^{-2}$).

How do you convert a speed given in $\text{km h}^{-1}$ into standard SI units of $\text{m s}^{-1}$?

Divide the speed in $\text{km h}^{-1}$ by $3.6$ (or multiply by $1000$ and divide by $3600$).

What is the fundamental difference between a scalar quantity and a vector quantity?

A scalar has magnitude (size) only, whereas a vector has both magnitude and direction.

What is the difference between distance and displacement, and between speed and velocity?

Distance and speed are scalars (total path length and rate of movement), while displacement and velocity are vectors (straight-line change in position with direction and rate of change of displacement with direction).

How do mass and weight differ in mechanics?

Mass is a scalar measuring the amount of matter in $\text{kg}$, while weight is a vector force measuring gravitational pull in $\text{N}$ ($W = mg$).

How are opposite directions represented in one-dimensional linear motion?

Opposite directions are represented using positive ($+$) and negative ($-$) signs.

How are 2D vectors represented using unit vectors $\mathbf{i}$ and $\mathbf{j}$?

A 2D vector is written as $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$, where $\mathbf{i}$ is a unit vector of length $1$ in the positive $x$-direction and $\mathbf{j}$ is a unit vector of length $1$ in the positive $y$-direction.

How do you calculate the magnitude and direction of a 2D vector $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$?

Magnitude is found using Pythagoras' theorem, $|\mathbf{r}| = \sqrt{x^2 + y^2}$, and direction is found using trigonometry, $\tan(\theta) = \frac{|\text{opposite}|}{|\text{adjacent}|}$.

How do you find the resultant of multiple force vectors given in $\mathbf{i}$-$\mathbf{j}$ notation?

Add the vectors together component-by-component by summing all the $\mathbf{i}$-components and all the $\mathbf{j}$-components separately.

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