Fundamental and derived quantities and their SI units, scalar and vector quantities, and the modelling assumptions used to simplify mechanics problems.

What this dot point is asking

Edexcel wants you to know the fundamental quantities (mass, length, time) and their SI units, derive units for quantities such as velocity, acceleration and force, distinguish scalar from vector quantities, and understand the modelling assumptions (such as treating a body as a particle or a string as light and inextensible) that simplify mechanics problems.

The answer

Fundamental and derived units

Many quantities are quoted in mixed units in real problems, so a routine first step is converting to SI. For example, a speed in $\text{km h}^{-1}$ is converted to $\text{m s}^{-1}$ by multiplying by $\dfrac{1000}{3600} = \dfrac{5}{18}$.

Deriving units

Every derived unit can be built from the base units by replacing each quantity with its unit. Velocity is displacement over time, so its unit is $\dfrac{\text{m}}{\text{s}} = \text{m s}^{-1}$. Acceleration is the rate of change of velocity, giving $\dfrac{\text{m s}^{-1}}{\text{s}} = \text{m s}^{-2}$. Force is mass times acceleration, so $\text{kg} \times \text{m s}^{-2} = \text{kg m s}^{-2}$, which is given the name newton. Checking that the units on each side of an equation match, called dimensional consistency, is a quick way to catch a slipped formula.

A short table of the common quantities helps fix the units. Displacement is in metres ($\text{m}$); velocity in $\text{m s}^{-1}$; acceleration in $\text{m s}^{-2}$; mass in kilograms ($\text{kg}$); force in newtons ($\text{N}$, or $\text{kg m s}^{-2}$); and weight, being a force, is also in newtons. Time is in seconds. Whenever a question quotes a quantity in a non-SI unit, such as a mass in grams or a distance in centimetres, converting to SI before substituting into any formula avoids a whole family of errors.

Scalars and vectors

A scalar quantity is fully described by its magnitude alone, such as distance, speed, mass and time. A vector quantity also needs a direction, such as displacement, velocity, acceleration and force. In mechanics, vectors are often written in $\mathbf{i}$, $\mathbf{j}$ component form. Adding two scalars is ordinary arithmetic, but adding two vectors must respect direction: two forces of $3$ N and $4$ N can combine to anything from $1$ N (opposing) up to $7$ N (aligned), and equal $5$ N when they act at right angles, by Pythagoras.

Modelling assumptions

Two more terms appear often. A rigid body keeps its shape and does not bend, which matters for beams and rods where turning effects are considered rather than treating everything as a single point. A uniform body has its mass spread evenly, so its weight acts at its geometric centre; this is why the weight of a uniform rod is placed at the midpoint. Each assumption removes a complication: ignoring air resistance makes the only force on a projectile its weight, a light string lets the tension be treated as the same throughout, and a smooth pulley lets a string pass over it without losing tension to friction.

Examples in context

Try this

Q1. State the SI unit of force and express it in base units. [2 marks]

• Cue. The newton, $1\,\text{N} = 1\,\text{kg m s}^{-2}$.

Q2. Classify each as scalar or vector: speed, displacement, mass, acceleration. [2 marks]

• Cue. Scalars: speed, mass. Vectors: displacement, acceleration.