Speed and Density

What you'll learn

How to use the formulas for average speed, density and pressure.
How to rearrange formulas to find a missing quantity.
How to convert mixed times like 4 hours 15 minutes correctly.
How to tackle whole-journey and liquid-mixture questions.

1. Compound measures: “per” means divide

Speed, density and pressure are all examples of compound measures. They compare two different quantities.

For example, miles per hour means “miles in 1 hour”, and grams per cm³ means “grams in 1 cm³”.

Definition

Compound measure

A compound measure is a measurement made from two other measurements, usually involving division. The word per often means “for each” or “divided by”.

The three key formulas are:

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}

Example

Finding pressure from force and area

A crate pushes down with a force of 96 N. The area touching the floor is 3 m². Find the pressure.

Choose the pressure formula.
$pressure=forcearea\text{pressure} = \frac{\text{force}}{\text{area}}$
Substitute the values.
$pressure=963=32\text{pressure} = \frac{96}{3} = 32$
The pressure is 32 N/m².

Tip

Units tell you the formula

If the answer unit is something like m/s, g/cm³ or N/m², it is usually a “divide” formula: first quantity divided by second quantity.

2. Average speed

Definition

Average speed

Average speed is the total distance divided by the total time. It does not mean the object travelled at exactly that speed for the whole journey.

The formula is:

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

You can also rearrange it:

distance=speed×timetime=distancespeed\begin{aligned} \text{distance} &= \text{speed} \times \text{time}\\ \text{time} &= \frac{\text{distance}}{\text{speed}} \end{aligned}

Basic speed questions

Example

Finding an average speed

A runner completes 160 metres in 20 seconds. Find the runner’s average speed.

Use the formula for average speed.
$speed=distancetime\text{average speed} = \frac{\text{distance}}{\text{time}}$
Substitute distance 160 metres and time 20 seconds.
$speed=16020=8\text{average speed} = \frac{160}{20} = 8$
The average speed is 8 m/s.

Converting time

This is where many marks are lost. If speed is in mph, the time must be in hours. If speed is in m/s, the time must be in seconds.

For example, 4 hours 15 minutes is not 4.15 hours. Since 15 minutes is a quarter of an hour, it is 4.25 hours.

Common Mistake

Using minutes as decimals

Do not write 3 hours 30 minutes as 3.30 hours. It is 3.5 hours, because 30 minutes is half an hour.

Example

Speed in miles per hour

A car travels 250 miles in 4 hours 30 minutes. Work out its average speed in mph, giving your answer to 1 decimal place.

Convert the time into hours.
$hours4\text{ hours }30\text{ minutes} = 4.5\text{ hours}$
Use average speed equals distance divided by time.
$speed=2504.5=55.555…\text{average speed} = \frac{250}{4.5} = 55.555\ldots$
Round to 1 decimal place: 55.6 mph.

Finding a time of arrival

When you are given distance and speed, first find the journey time, then add it to the starting time.

Example

Finding an arrival time

Nadia leaves home at 08:10. She drives 63 miles at an average speed of 28 mph. Find her arrival time.

Use the time formula.
$time=distancespeed\text{time} = \frac{\text{distance}}{\text{speed}}$
Substitute the values.
$hours\text{time} = \frac{63}{28} = 2.25\text{ hours}$
Convert 2.25 hours into hours and minutes: 2 hours 15 minutes.
Add 2 hours 15 minutes to 08:10.
Nadia arrives at 10:25.

3. Average speed over a whole journey

For a whole journey, always use:

time\text{average speed for whole journey} = \frac{\text{total distance}}{\text{total time}}

Do not just average the separate speeds unless the times are the same and the question genuinely allows it.

Common Mistake

Averaging the speeds

If one part of a journey is at 60 mph and another part is at 40 mph, the whole journey speed is not automatically 50 mph. Use total distance divided by total time.

Example

A journey in two parts

Mina drives 310 miles. She drives the first 150 miles at 60 mph. The rest of the journey takes 3 hours 20 minutes. Find her average speed for the whole journey.

Find the time for the first part.
$hours\text{time} = \frac{150}{60} = 2.5\text{ hours}$
Convert the second time into hours.
$hours3\text{ hours }20\text{ minutes} = 3 + \frac{20}{60} = 3.333\ldots\text{ hours}$
Add the times.
$hours\text{total time} = 2.5 + 3.333\ldots = 5.833\ldots\text{ hours}$
Divide total distance by total time.
$speed=3105.833…=53.142…\text{average speed} = \frac{310}{5.833\ldots} = 53.142\ldots$
The average speed is about 53.1 mph.

Tip

Same journey means same distance

If you use one person’s speed and time to find a distance, you are assuming another person travelled the same distance. If they took a different route, the distance might be different.

Assuming the same speed

Some questions ask you to assume someone keeps the same speed for a longer distance. This is a proportional reasoning question: if the speed stays constant, time and distance increase in the same ratio.

Example

Running further at the same speed

Elliot runs 3.2 miles in 15 minutes 20 seconds. Assuming he keeps the same speed, estimate how long it would take him to run 8 miles.

Convert 15 minutes 20 seconds into seconds.
$15 \times 60 + 20 = 920\text{ seconds}$
Find the time for 1 mile.
$mile\frac{920}{3.2} = 287.5\text{ seconds per mile}$
Multiply by 8 miles.
$287.5 \times 8 = 2300\text{ seconds}$
Convert 2300 seconds into minutes and seconds: 38 minutes 20 seconds.

Common Mistake

Real-life speed changes

If a runner slows down as the distance increases, the calculated time using the same speed will be too small. The real time would be longer.

4. Density

Definition

Density

Density is mass per unit volume. It tells you how much mass is packed into each 1 cm³ or each 1 ml.

The main formula is:

density=massvolume\text{density} = \frac{\text{mass}}{\text{volume}}

Rearranged forms:

mass=density×volumevolume=massdensity\begin{aligned} \text{mass} &= \text{density} \times \text{volume}\\ \text{volume} &= \frac{\text{mass}}{\text{density}} \end{aligned}

Finding density

Example

Density of a solid

A piece of metal has mass 540 g and volume 30 cm³. Find its density.

Use density equals mass divided by volume.
$density=massvolume\text{density} = \frac{\text{mass}}{\text{volume}}$
Substitute the values.
$density=54030=18\text{density} = \frac{540}{30} = 18$
The density is 18 g/cm³.

Finding mass or volume

Example

Finding volume from mass and density

A stone has mass 72 g and density 4.5 g/cm³. Find its volume.

Use the rearranged formula for volume.
$volume=massdensity\text{volume} = \frac{\text{mass}}{\text{density}}$
Substitute the values.
$volume=724.5=16\text{volume} = \frac{72}{4.5} = 16$
The volume is 16 cm³.

Example

Finding mass of a liquid

A liquid has density 1.2 g/ml. Find the mass of 300 ml of the liquid.

Use mass equals density multiplied by volume.
$mass=density×volume\text{mass} = \text{density} \times \text{volume}$
Substitute the values.
$mass=1.2×300=360\text{mass} = 1.2 \times 300 = 360$
The mass is 360 g.

5. Mixtures and density

When liquids are mixed, track mass and volume separately. The masses add together, and the volumes add together, then you calculate the final density.

Diagram showing two liquids being mixed, with masses and volumes adding to form a final mixture

Key Idea

Mixture method

For mixtures, do not combine densities directly. First find total mass and total volume, then use density equals total mass divided by total volume.

Common Mistake

Adding densities

You cannot add densities like 0.8 g/ml plus 1.2 g/ml to get the density of the mixture. Density depends on both mass and volume.

Example

Finding the density of an unknown liquid in a mixture

Liquid A has density 1.4 g/cm³ and volume 120 cm³. It is mixed with some Liquid B. The final mixture has mass 288 g and density 1.2 g/cm³. Find the density of Liquid B.

Find the mass of Liquid A.
$A=1.4×120=168\text{mass of A} = 1.4 \times 120 = 168$
Find the total volume of the final mixture.
$mixture=2881.2=240\text{volume of mixture} = \frac{288}{1.2} = 240$
Liquid B has volume 240 - 120 = 120 cm³.
Liquid B has mass 288 - 168 = 120 g.
Find the density of Liquid B.
$B=120120=1\text{density of B} = \frac{120}{120} = 1$
The density of Liquid B is 1 g/cm³.

Example

Finding the density of a final mixture

150 ml of Liquid X has density 0.8 g/ml. 250 ml of Liquid Y has density 1.2 g/ml. They are mixed together. Find the density of the mixture.

Find the mass of Liquid X.
$0.8 \times 150 = 120$
Find the mass of Liquid Y.
$1.2 \times 250 = 300$
Add the masses and volumes.
$volume=150+250=400\begin{aligned} \text{total mass} &= 120 + 300 = 420\\ \text{total volume} &= 150 + 250 = 400 \end{aligned}$
Find the density of the mixture.
$density=420400=1.05\text{density} = \frac{420}{400} = 1.05$
The density of the mixture is 1.05 g/ml.

Exam technique

In the exam

Write the correct formula first, then identify which quantity is missing.
Convert time before substituting: use hours for mph and seconds for m/s.
For whole journeys and mixtures, use totals: total distance divided by total time, or total mass divided by total volume.
Keep units in your final answer, and round only at the end unless the question tells you otherwise.

Self review

Check yourself

If a speed is given in mph, what unit should the time be in?
How do you find volume when you know mass and density?
In a mixture question, why should you calculate total mass and total volume before finding density?

Speed and Density

What you'll learn

How to use the formulas for average speed, density and pressure.
How to rearrange formulas to find a missing quantity.
How to convert mixed times like 4 hours 15 minutes correctly.
How to tackle whole-journey and liquid-mixture questions.

1. Compound measures: “per” means divide

Speed, density and pressure are all examples of compound measures. They compare two different quantities.

For example, miles per hour means “miles in 1 hour”, and grams per cm³ means “grams in 1 cm³”.

Definition

Compound measure

A compound measure is a measurement made from two other measurements, usually involving division. The word per often means “for each” or “divided by”.

The three key formulas are:

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}

Example

Finding pressure from force and area

A crate pushes down with a force of 96 N. The area touching the floor is 3 m². Find the pressure.

Choose the pressure formula.
$pressure=forcearea\text{pressure} = \frac{\text{force}}{\text{area}}$
Substitute the values.
$pressure=963=32\text{pressure} = \frac{96}{3} = 32$
The pressure is 32 N/m².

Tip

Units tell you the formula

If the answer unit is something like m/s, g/cm³ or N/m², it is usually a “divide” formula: first quantity divided by second quantity.