Speed and Density

What you'll learn

Choose and use formulas for speed, density and pressure.
Rearrange formulas to find missing quantities.
Convert time units for mph and m/s questions.
Combine masses and volumes in liquid-mixture problems.

The basic idea: “per” means divide

A unit is the label on a measurement, such as metres, seconds, grams or cm³.

You will meet these quantities:

Distance: how far something travels.
Time: how long something takes.
Mass: how much matter there is, often measured in grams or kilograms.
Volume: how much 3D space something takes up, often measured in cm³ or ml.
Force: a push or pull, measured in newtons (N).
Area: the size of a surface, such as the bottom of a box.

The word per means “for each one”. So miles per hour means miles divided by hours.

Definition

The three rates

Average speed is distance per unit time.
Density is mass per unit volume.
Pressure is force per unit area.

The three main formulae are:

The three rate formulae all have the same ‘amount divided by per quantity’ structure.

speed=distancetimedensity=massvolumepressure=forcearea\begin{aligned} \text{average speed} &= \frac{\text{distance}}{\text{time}} \\ \text{density} &= \frac{\text{mass}}{\text{volume}} \\ \text{pressure} &= \frac{\text{force}}{\text{area}} \end{aligned}

Key Idea

One pattern

Each formula compares one quantity with another by dividing. If you know two quantities, you can usually find the third.

Worked example: average speed

Example

Finding average speed

A runner covers 240 metres in 30 seconds. Find the average speed.

The runner’s average speed is found from the total distance and total time.

Write the formula.
$speed=distancetime\text{average speed} = \frac{\text{distance}}{\text{time}}$
Substitute the distance and time.
$speed=24030=8\text{average speed} = \frac{240}{30} = 8$
Add the correct units: the average speed is 8 m/s.

Density and pressure: same pattern

To substitute means to replace the words in a formula with the numbers from the question.

For density, divide mass by volume.
For pressure, divide force by area.

Example

Quick density and pressure questions

A metal object has mass 630 g and volume 35 cm³. Use the density formula.

Density compares the mass of an object with the volume it takes up.

$$
\text{density} = \frac{630}{35} = 18
$$

2. The density is 18 g/cm³.

A crate exerts a force of 150 N over an area of 6 m². Use the pressure formula.

Pressure depends on the force spread over the contact area.

$$
\text{pressure} = \frac{150}{6} = 25
$$

4. The pressure is 25 N/m².

Rearranging when the missing value is not on top

Sometimes the question asks for distance, time, mass, volume, force or area instead.

For a formula like $a=bca=\frac{b}{c}$ :

The top quantity is found by multiplying: $\times c$ .
The bottom quantity is found by dividing: $c=bac=\frac{b}{a}$ .

Tip

Cover-up method

In formulas shaped like “top divided by bottom”, cover the missing quantity. If the missing quantity is on top, multiply. If it is on the bottom, divide.

Example

Finding volume and area

A stone has mass 84 g and density 3.5 g/cm³. Volume is the bottom quantity, so divide mass by density.

When mass and density are known, the missing volume is found by division.

$$
\text{volume} = \frac{84}{3.5} = 24
$$

2. The volume is 24 cm³.

A block has force 108 N and pressure 36 N/m². Area is the bottom quantity, so divide force by pressure.
$area=10836=3\text{area} = \frac{108}{36} = 3$
The area is 3 m².

Time units in speed questions

For a speed in miles per hour, the time must be in hours.
For a speed in metres per second, the time must be in seconds.

To convert minutes to hours, divide by 60.
To convert seconds to minutes, divide by 60.

Common Mistake

Writing minutes as decimals incorrectly

Do not write 4 hours 15 minutes as 4.15 hours. Since 15 minutes is $1560=0.25\frac{15}{60}=0.25$ hours, the correct time is 4.25 hours.

Example

Speed in mph and arrival time

A car travels 310 miles in 5 hours 40 minutes. Convert the time into hours.

For miles per hour, the journey time must be converted into hours before calculating speed.

$$
5+\frac{40}{60}=5.666\ldots
$$

2. Use average speed equals distance divided by time.

$$
\text{average speed}=\frac{310}{5.666\ldots}=54.705\ldots
$$

3. To 1 decimal place, the average speed is 54.7 mph.

For a separate journey, a driver travels 81 miles at 36 mph. Find the time.
$time=8136=2.25\text{time}=\frac{81}{36}=2.25$
2.25 hours is 2 hours 15 minutes, so if the driver leaves at 08:20, they arrive at 10:35.

Average speed for a whole journey

For a whole journey, use:

time\text{average speed}=\frac{\text{total distance}}{\text{total time}}

This is true even if the speed changes during the journey.

Common Mistake

Averaging the speeds

If a journey has two parts, do not usually add the two speeds and divide by 2. Use total distance divided by total time instead.

Example

Average speed over two parts

A van travels 150 miles at 60 mph. It then travels another 90 miles in 2 hours 15 minutes. Find the average speed for the whole journey.

For a whole journey, add the distances and times before finding the average speed.

Find the time for the first part.
$time=15060=2.5\text{time}=\frac{150}{60}=2.5$
Convert the second time into hours.
$2+1560=2.252+\frac{15}{60}=2.25$
Add the distances and times.
$time=2.5+2.25=4.75\begin{aligned} \text{total distance} &= 150+90=240 \\ \text{total time} &= 2.5+2.25=4.75 \end{aligned}$
Find the average speed.
$speed=2404.75=50.526…\text{average speed}=\frac{240}{4.75}=50.526\ldots$
The average speed is about 50.5 mph.

Common Mistake

Different routes

If two people take different roads, the distance may be different. You can only use the same distance if the question makes it clear they travelled the same journey.

Mixing liquids and density

A mixture is made by combining substances. In these questions, work with mass and volume separately.

Do not add densities directly. Density is found after you know the total mass and total volume.

Tip

Mixtures

For mixed liquids: find masses, find volumes, then use density equals total mass divided by total volume.

Example

Finding the density of an unknown liquid

120 cm³ of Liquid A has density 1.25 g/cm³. It is mixed with some Liquid B to make Liquid C. Liquid C has mass 240 g and density 1.2 g/cm³. Find the density of Liquid B.

In mixture problems, track mass and volume separately for each liquid.

Find the total volume of Liquid C.
$C=2401.2=200\text{volume of C}=\frac{240}{1.2}=200$
Find the mass of Liquid A.
$A=1.25×120=150\text{mass of A}=1.25 \times 120=150$
Subtract to find the volume and mass of Liquid B.
$B=240−150=90\begin{aligned} \text{volume of B} &= 200-120=80 \\ \text{mass of B} &= 240-150=90 \end{aligned}$
Find the density of Liquid B.
$B=9080=1.125\text{density of B}=\frac{90}{80}=1.125$
The density of Liquid B is 1.125 g/cm³.

Exam technique

In the exam

Write the formula first, then substitute the numbers.
Check the units before calculating, especially hours, minutes and seconds.
For whole journeys and mixtures, use totals: total distance, total time, total mass and total volume.

Self review

Check yourself

If a car travels 150 miles in 2.5 hours, what calculation gives its average speed?
Why is 3 hours 20 minutes not the same as 3.20 hours?
In a liquid mixture question, why should you avoid adding the densities directly?

Speed and Density

What you'll learn

Choose and use formulas for speed, density and pressure.
Rearrange formulas to find missing quantities.
Convert time units for mph and m/s questions.
Combine masses and volumes in liquid-mixture problems.

The basic idea: “per” means divide

A unit is the label on a measurement, such as metres, seconds, grams or cm³.

You will meet these quantities:

Distance: how far something travels.
Time: how long something takes.
Mass: how much matter there is, often measured in grams or kilograms.
Volume: how much 3D space something takes up, often measured in cm³ or ml.
Force: a push or pull, measured in newtons (N).
Area: the size of a surface, such as the bottom of a box.

The word per means “for each one”. So miles per hour means miles divided by hours.

Definition

The three rates

Average speed is distance per unit time.
Density is mass per unit volume.
Pressure is force per unit area.

The three main formulae are:

The three rate formulae all have the same ‘amount divided by per quantity’ structure.

speed=distancetimedensity=massvolumepressure=forcearea\begin{aligned} \text{average speed} &= \frac{\text{distance}}{\text{time}} \\ \text{density} &= \frac{\text{mass}}{\text{volume}} \\ \text{pressure} &= \frac{\text{force}}{\text{area}} \end{aligned}

Key Idea

One pattern

Each formula compares one quantity with another by dividing. If you know two quantities, you can usually find the third.

Worked example: average speed

Example

Finding average speed

A runner covers 240 metres in 30 seconds. Find the average speed.

The runner’s average speed is found from the total distance and total time.

Write the formula.
$speed=distancetime\text{average speed} = \frac{\text{distance}}{\text{time}}$
Substitute the distance and time.
$speed=24030=8\text{average speed} = \frac{240}{30} = 8$
Add the correct units: the average speed is 8 m/s.

Density and pressure: same pattern

To substitute means to replace the words in a formula with the numbers from the question.

For density, divide mass by volume.
For pressure, divide force by area.

Example

Quick density and pressure questions

A metal object has mass 630 g and volume 35 cm³. Use the density formula.

Density compares the mass of an object with the volume it takes up.

$$
\text{density} = \frac{630}{35} = 18
$$

2. The density is 18 g/cm³.

A crate exerts a force of 150 N over an area of 6 m². Use the pressure formula.

Pressure depends on the force spread over the contact area.

$$
\text{pressure} = \frac{150}{6} = 25
$$

4. The pressure is 25 N/m².

Rearranging when the missing value is not on top

Sometimes the question asks for distance, time, mass, volume, force or area instead.

For a formula like $a=bca=\frac{b}{c}$ :

The top quantity is found by multiplying: $\times c$ .
The bottom quantity is found by dividing: $c=bac=\frac{b}{a}$ .

Tip

Cover-up method

In formulas shaped like “top divided by bottom”, cover the missing quantity. If the missing quantity is on top, multiply. If it is on the bottom, divide.

Example

Finding volume and area

A stone has mass 84 g and density 3.5 g/cm³. Volume is the bottom quantity, so divide mass by density.

When mass and density are known, the missing volume is found by division.

$$
\text{volume} = \frac{84}{3.5} = 24
$$

2. The volume is 24 cm³.

A block has force 108 N and pressure 36 N/m². Area is the bottom quantity, so divide force by pressure.
$area=10836=3\text{area} = \frac{108}{36} = 3$
The area is 3 m².

Time units in speed questions

For a speed in miles per hour, the time must be in hours.
For a speed in metres per second, the time must be in seconds.

To convert minutes to hours, divide by 60.
To convert seconds to minutes, divide by 60.

Common Mistake

Writing minutes as decimals incorrectly

Do not write 4 hours 15 minutes as 4.15 hours. Since 15 minutes is $1560=0.25\frac{15}{60}=0.25$ hours, the correct time is 4.25 hours.

Example

Speed in mph and arrival time

A car travels 310 miles in 5 hours 40 minutes. Convert the time into hours.

For miles per hour, the journey time must be converted into hours before calculating speed.

$$
5+\frac{40}{60}=5.666\ldots
$$

2. Use average speed equals distance divided by time.

$$
\text{average speed}=\frac{310}{5.666\ldots}=54.705\ldots
$$

3. To 1 decimal place, the average speed is 54.7 mph.

For a separate journey, a driver travels 81 miles at 36 mph. Find the time.
$time=8136=2.25\text{time}=\frac{81}{36}=2.25$
2.25 hours is 2 hours 15 minutes, so if the driver leaves at 08:20, they arrive at 10:35.

Average speed for a whole journey

For a whole journey, use:

time\text{average speed}=\frac{\text{total distance}}{\text{total time}}

This is true even if the speed changes during the journey.

Common Mistake

Averaging the speeds

If a journey has two parts, do not usually add the two speeds and divide by 2. Use total distance divided by total time instead.

Example

Average speed over two parts

A van travels 150 miles at 60 mph. It then travels another 90 miles in 2 hours 15 minutes. Find the average speed for the whole journey.

For a whole journey, add the distances and times before finding the average speed.

Find the time for the first part.
$time=15060=2.5\text{time}=\frac{150}{60}=2.5$
Convert the second time into hours.
$2+1560=2.252+\frac{15}{60}=2.25$
Add the distances and times.
$time=2.5+2.25=4.75\begin{aligned} \text{total distance} &= 150+90=240 \\ \text{total time} &= 2.5+2.25=4.75 \end{aligned}$
Find the average speed.
$speed=2404.75=50.526…\text{average speed}=\frac{240}{4.75}=50.526\ldots$
The average speed is about 50.5 mph.

Common Mistake

Different routes

If two people take different roads, the distance may be different. You can only use the same distance if the question makes it clear they travelled the same journey.

Mixing liquids and density

A mixture is made by combining substances. In these questions, work with mass and volume separately.

Do not add densities directly. Density is found after you know the total mass and total volume.

Tip

Mixtures

For mixed liquids: find masses, find volumes, then use density equals total mass divided by total volume.

Example

Finding the density of an unknown liquid

120 cm³ of Liquid A has density 1.25 g/cm³. It is mixed with some Liquid B to make Liquid C. Liquid C has mass 240 g and density 1.2 g/cm³. Find the density of Liquid B.

In mixture problems, track mass and volume separately for each liquid.

Find the total volume of Liquid C.
$C=2401.2=200\text{volume of C}=\frac{240}{1.2}=200$
Find the mass of Liquid A.
$A=1.25×120=150\text{mass of A}=1.25 \times 120=150$
Subtract to find the volume and mass of Liquid B.
$B=240−150=90\begin{aligned} \text{volume of B} &= 200-120=80 \\ \text{mass of B} &= 240-150=90 \end{aligned}$
Find the density of Liquid B.
$B=9080=1.125\text{density of B}=\frac{90}{80}=1.125$
The density of Liquid B is 1.125 g/cm³.

Exam technique

In the exam

Write the formula first, then substitute the numbers.
Check the units before calculating, especially hours, minutes and seconds.
For whole journeys and mixtures, use totals: total distance, total time, total mass and total volume.

Self review

Check yourself

If a car travels 150 miles in 2.5 hours, what calculation gives its average speed?
Why is 3 hours 20 minutes not the same as 3.20 hours?
In a liquid mixture question, why should you avoid adding the densities directly?

Speed and Density

What you'll learn

The basic idea: “per” means divide

Worked example: average speed

Density and pressure: same pattern

Rearranging when the missing value is not on top

Time units in speed questions

Average speed for a whole journey

Mixing liquids and density

Recap questions

Speed and Density

What you'll learn

The basic idea: “per” means divide

Worked example: average speed

Density and pressure: same pattern

Rearranging when the missing value is not on top

Time units in speed questions

Average speed for a whole journey

Mixing liquids and density

Recap questions

Speed and Density

What you'll learn

The basic idea: “per” means divide

Worked example: average speed

Density and pressure: same pattern

Rearranging when the missing value is not on top

Time units in speed questions

Average speed for a whole journey

Mixing liquids and density

Recap questions

Speed and Density Revision Guide

Speed and Density

What you'll learn

The basic idea: “per” means divide

Worked example: average speed

Density and pressure: same pattern

Rearranging when the missing value is not on top

Time units in speed questions

Average speed for a whole journey

Mixing liquids and density

Recap questions

Speed and Density Revision Guide