Speed, density and pressure: Formulae and units

In a nutshell

Speed, density and pressure are three quantities used all the time in the real world. They each have their own formula which can be rearranged to calculate other quantities.

Definitions

	Definition	Formula triangle	Units
Speed	The distance travelled per unit time.		$m/s$ - metres per second $km/h$ - kilometres per hour
Density	The mass per unit volume of a substance.		$kg/m^3$ - kilograms per metre cubed $g/cm^3$ - grams per centimetre cubed
Pressure	The force acting per unit area.		$N/m^2$ - newtons per metre squared $Pa$ - pascals

Note: Newtons are used to measure force!

Formula triangles

Formula triangles are really useful tools to help rearrange formulae. To use them, simply cover the quantity that you want to find out and write down whatever is left as the formula.

Example 1

Find the formulae for a, b and c.

Cover each letter and write down the remaining formula.

$\begin {aligned}a&=\frac{b}{c}\\b&=a\times c\\c&=\frac{b}{a}\end {aligned}$

Maths; Ratio proportion and rates of change; KS4 Year 10; Speed, density and pressure: Formulae and units

Speed, density and mass units

All three quantities are made up of two units of measure and therefore conversions between units need to be done in steps. If two conversions are needed, convert one of the two units first and then the other.

Note: It's vital that units remain consistent across each of the three measures used in the equation.

Example 2

Convert $150,000,000 \enspace m/s$ into $km/h$ . Give your answer to one decimal place.

Start by converting into kilometres.

$1km=1000m$

$150,000,000\div1000=150,000$

Then convert into minutes and then hours.

$1\text{ min} = 60 \text{ secs}\\ 1 \text{ hour} = 60 \text{ mins}\\150,000\div60= 2500 \\2500\div60 = 41.6666 = \underline{41.7km/h}$

Speed

Using the formula triangle, the formula for speed can be rearranged in three ways where: S = speed, D = distance and T = time.

$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$

$\text{Distance}=\text{Speed}\times\text{Time}$

$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

Example 3

Anna is having a race with her sister between two lamp posts which are $25$ metres apart. It took Anna $6.8$ seconds to run between the two - what speed was she going at? Give your answer to two decimal places.

Use the formula triangle to write down the formula for speed.

$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$ 

Plug the values into the formula.

$\text{Speed}=\frac{25}{6.8}=3.67647059=3.68$ to two decimal places.

Look back to the question to find units and add these to the answer. Since metres and seconds have been used, the units are metres per second: $m/s$ .

$\underline{3.68m/s}$

Density

Using the formula triangle, the formula for density can also be rearranged in three ways where: D = density, M = mass and V = volume.

$\text{Density}=\frac{\text{Mass}}{\text{Volume}}$

$\text{Mass}=\text{Density}\times\text{Volume}$

$\text{Volume}=\frac{\text{Mass}}{\text{Density}}$

Example 4

A solid chocolate egg has a mass of $320g$ . If its density is $3.4g/cm^3$ , what is the volume of the egg? Give your answer to two decimal places.

Use the formula triangle to write down the formula for volume.

$\text{Volume}=\frac{\text{Mass}}{\text{Density}}$ 

Plug in the values into the formula.

$\text{Volume} =\frac{320}{3.4}=94.1176471=94.12$ to two decimal places.

Select units based on what is used in the question.

$\underline{94.12cm^3}$

Pressure

Using the formula triangle once more, the formula for speed can be rearranged in three ways where: P = pressure, F = force, and A = area.

$\text{Pressure}=\frac{\text{Force}}{\text{Area}}$

$\text{Force }= \text{Pressure} \times \text{Area}$

$\text{Area}=\frac{\text{Force}}{\text{Pressure}}$

Example 5

A $50cm\times30cm$ book exerts a pressure of $12 N/m^2$ onto a table top. Calculate the force exerted by the book.

Use the formula triangle to write down the formula for force.

$\text{Force }= \text{Pressure} \times \text{Area}$

Calculate the area of the book.

$50cm\times30cm=1500cm^2$

Check the units in the question. Pressure is measure in $N/m^2$ so $cm^2$ must be converted to $m^2$ .

$1500\div100\div100=0.15m^2$

Plug into the equation.

$\text{Force}=12\times0.15=1.8$

Add units.

$\underline{1.8N}$