Compound measures: speed, distance and time; density, mass and volume; pressure, force and area; and converting between compound units such as metres per second and kilometres per hour.

What this dot point is asking

A compound measure combines two units, such as kilometres per hour or grams per cubic centimetre. Edexcel expects you to use the relationships between speed, distance and time; density, mass and volume; and pressure, force and area; and to convert between compound units. These appear in real-world problems, and the formula triangles make the three rearrangements easy to recall.

Speed, distance and time

Speed measures how fast something moves: the distance covered per unit of time.

The most frequent difficulty is time. Minutes must be written as a fraction of an hour for km/h: $15$ minutes is $\tfrac{15}{60} = 0.25$ hours, and $45$ minutes is $0.75$ hours. Average speed over a whole journey is the total distance divided by the total time, not the average of separate speeds.

Density, mass and volume

Density measures how much mass is packed into a given volume.

So a block of mass $600\,\text{g}$ and volume $75\,\text{cm}^3$ has density $\dfrac{600}{75} = 8\,\text{g/cm}^3$. To find the mass of a $50\,\text{cm}^3$ piece of the same material, use mass $=$ density $\times$ volume $= 8 \times 50 = 400\,\text{g}$.

Pressure, force and area

Pressure measures how concentrated a force is over an area.

A force of $200\,\text{N}$ on an area of $4\,\text{m}^2$ gives a pressure of $\dfrac{200}{4} = 50\,\text{N/m}^2$. The same force on a smaller area gives a higher pressure, which is why a sharp knife or a drawing pin works.

Converting compound units

Converting a compound unit requires changing both parts.

Why the units carry the meaning

Compound measures are best understood by reading their units as a sentence. "Kilometres per hour" literally means kilometres travelled in each hour, "grams per cubic centimetre" means grams of mass in each cubic centimetre, and "newtons per square metre" means newtons of force on each square metre. This is why writing the formula first and tracking the units protects you from rearranging the wrong way: if an answer should be in km/h but your working gives hours per km, you have divided the wrong pair. In multi-step problems, such as finding how long a journey takes at a given speed, the units also tell you which quantity to solve for.

Try this

Q1. A runner covers $400\,\text{m}$ in $50$ seconds. Work out the average speed in metres per second. [2 marks]

• Cue. Speed $= \dfrac{400}{50} = 8\,\text{m/s}$.

Q2. Gold has a density of $19.3\,\text{g/cm}^3$. Work out the mass of a $5\,\text{cm}^3$ gold bar. [2 marks]

• Cue. Mass $=$ density $\times$ volume $= 19.3 \times 5 = 96.5\,\text{g}$.