GCSE Maths Ratio and Proportion

Speed Distance Time

Formula for Speed

# Formula For Speed

Here we will learn about the formula for speed including understanding and using the terms constant speed and average speed. We will also be calculating the average speed of an object given its distance and time. This will extend to using and applying the speed formula and solving problems involving the formula for speed.

There are also worksheets on the formula for speed distance and time based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is the formula for speed?

The formula for speed is given by

Speed \textbf{=} distance \textbf{÷} time

“Speed equals distance divided by time”

You will normally shorten this to the below:

The speed of an object is travelling at is related to the distance it is covering in relation to how long it was travelling for (time).

### What is the formula for speed? ### What is speed?

Speed is about how fast an object moves. The speed of the moving object is found by calculating the relationship between the distance the object travels and the period of time taken to travel the distance.

Some examples of the units of speed are:

• Metres per second (m/s)
• Miles per hour (mph)
• Km per hour (km/h)
• Speed of sound - 343 metres per second
• Speed of light - 299 \ 792 \ 458 metres per second

Constant speed is where the speed does not change, e.g. a straight line on a distance-time graph indicates a constant speed.

We can calculate the average speed of an object by dividing the total distance by the total time.

If you also consider the direction of the movement as well at its speed then this is called the object’s velocity.

### What is time?

Time can be defined as the ongoing sequence of events taking place.

Some examples of the units of time are:

• Seconds (sec)
• Minutes (mins)
• Hours (hrs)
• Days

Note: The SI unit (Standard Unit Measurement) for time is a second.

### What is distance?

Distance is the length of space between two points.

Some examples of the units of distance are:

• Millimetres (mm)
• Centimetres (cm)
• Metres (m)
• Kilometres (km)
• Miles

Note: if we only consider how far the object has moved in regards to its starting point this is called displacement.

### Rate of change

The speed of an object is the rate of change between the distance covered and the amount of time taken. When working with graphs the rate of change can be found by calculating the gradient of the distance-time graph.

Step-by-step guide: Speed distance time (coming soon)

## How to use the formula for speed

In order to use the formula for speed:

1. Write down which variable you know with their units.
2. Check the units align, do any need to be converted?
3. Write down the formula for speed.
4. Solve to find speed (or distance or time).

### Explain how to use the formula for speed ## Formula for speed examples

### Example 1: calculating speed by substituting values into the speed formula

Question: A car travels at a constant speed. The train travels 100 \ miles in 2 \ hours.

1. Write down which variable you know with their units.

Speed: unknown

Distance: 100 \ miles

Time: 2 \ hours

2Check the units align, do any need to be converted?

Units are correct as the questions’ units are already in miles and hours.

3Write down the formula for speed.

\text{Speed } = \text{ distance } \div \text{ time}

4Solve to find speed (or distance or time).

We are finding the speed.

\begin{aligned} \text{Speed} &= \text{distance}\div \text{time} \\\\ \text{Speed} &= 100\ \text{miles}\div 2\ \text{hours} \\\\ \text{Speed} &= 50\ \text{miles per hour} \ \text{(mph)} \end{aligned}

The car travels at 50 \ miles \ per \ hour \ (mph)

### Example 2: calculating speed by substituting values into the speed formula

Question: A train travels at a constant speed. The train travels a total distance of 450 \ km in a period of time of 4 \ hours.

Speed: unknown

Distance: 450 \ km

Time: 4 \ hours

Units are correct as the questions’ units are already in miles and hours.

\text{Speed } = \text{ distance } \div \text{ time}

We are finding the speed.

\begin{aligned} \text{Speed} &= \text{distance}\div \text{time} \\\\ \text{Speed} &= 450\ \text{km}\div 4\ \text{hours} \\\\ \text{Speed} &= 112.5 kmph \end{aligned}

The car travels at 112.5 \ km/h

### Example 3: calculating speed by substituting values into the speed formula with unit conversion

Question: A person walks at a constant speed. They travel 600 \ m in 40.25 \ hours.

Speed: unknown

Distance: 600 \ m

Time: 0.25 \ hours

The distance is already given in metres so you do not need to convert this unit.

The time is given in hours so you need to convert this to minutes.

0.25 \ hours \times 60=15 \ minutes

Speed: unknown

Distance: 600 \ m

Time: 15 \ minutes

\text{Speed } = \text{ distance } \div \text{ time}

We are finding the speed.

\begin{aligned} \text{Speed} &= \text{distance}\div \text{time} \\\\ \text{Speed} &= 600\ \text{m}\div 15\ \text{minutes} \\\\ \text{Speed} &= 40 \ m/min \end{aligned}

The car travels at 40 \ m/min

### Example 4: using speed to find another value

Question: Phil is in a running race. Phil runs at an average speed of 5 \ m/s.

The race is 21 \ km long.

What is the amount of time Phil takes to finish the race?

Speed: 5 \ m/s

Distance: 21 \ km

Time: unknown

The question used two different units of distance, you need to convert one so you are working with equivalent distances. In this question it is easier to convert 20 \ km to m.

21 \ km \times 100=21000 \ m

Because you now have equivalent units of distances you can calculate Phil’s time before converting to hours and minutes.

\text{Speed } = \text{ distance } \div \text{ time}

We are finding the time. We need to adapt the speed formula.

\begin{aligned} \text {Time }&=\text { distance } \div \text { speed } \\\\ \text {Time }&=21000 m \div 5 \ m/ s \\\\ \text {Time }&=4200 \text { seconds } \end{aligned}

You have calculated the time to be 4200 \ seconds.

However the question wants the time in hours and minutes,

Remember: seconds \div 60 = minutes

4200 \div 60=70 \ minutes

Therefore

70 \ minutes=1 \ hr and 10 mins

Therefore

Phil completes the race in 1 \ hour and 10 \ minutes

### Example 5: using speed to find another value

Question: A train travels at 120 \ km/h for 3\frac{1}{4} \ hours. Calculate the total distance travelled.

Speed: 120 \ km/h

Distance: unknown

Time: 3\frac{1}{4} \ hours=3.25 \ hours

The units align. Sometime people prefer decimals to fractions, so we can use 3\frac{1}{4} or 3.25

\text{Speed } = \text{ distance } \div \text{ time}

We are finding the time. We need to adapt the speed formula.

\begin{aligned} \text {Distance }&=\text {speed} \times \text {time} \\\\ \text {Distance }&=120 \ \text{km/h} \times 3.25\ \text{hours} \\\\ \text {Distance }&=390 \text { km } \end{aligned}

You have calculated the time to be 390 \ kilometres.

Therefore the distance of the journey is 390 \ km.

### Example 6: problem involving speed

Miss Yellow completes a journey in 3 stages.

In the first stage she travels at 33.75 \ km in 45 \ minutes.

In the second stage she travels 258 \ km in 4 \ hours and 18 \ minutes.

The final stage is much shorter and she travels 1200 \ m in 90 \ seconds.

What was her average speed for the whole journey?

You want to know Miss Yellow’s overall average speed so you need to calculate her total distance and total time

Speed: unknown

Distance: 33.75 \ km + 258 \ km + 1200 \ m

Time: 45 \ minutes + 4 \ hours and 18 \ minutes + 90 \ Seconds

You will notice in the above the units are not the same for distance or speed. You need to convert them all first before proceeding.

Distance: convert all to km

1200 \ m = 1.2 \ km

Therefore total distance is

33.75 \ km + 258 \ km + 1.2 \ km = 292.96

Time: convert all to minutes

4 \ hours and 18 \ minutes = 258 \ minutes

90 \ seconds = 1.5 \ minutes

Therefore total time is

45 \ minutes + 258 \ minutes + 1.5 \ minutes =304.5 \ minutes

\text{Speed } = \text{ distance } \div \text{ time}

\begin{aligned} \text{Speed} &= \text{distance}\div \text{time} \\\\ \text{Speed} &= 292.96 \ \text{km}\div 305.5 \ \text{minutes} \\\\ \text{Speed} &= 0.95895... km/min \end{aligned}

0.95895... \ km/min needs to be rounded to 3 significant figures

The car travels at 0.959 \ km/min

### Common misconceptions

• Incorrect formula for speed

You must remember the speed formula with the correct operations between distance and time.

\text{Speed } = \text{ distance } \div \text{ time}

• Units

You must remember the relationship between the units is important for the context of the question. If you are given an object’s speed in km per hour and the time as minutes you need to first convert one of the units to be able to give an answer in hours.

E.g. km/h \div minutes does not give you km

• Check if you answer is ‘sensible’

If you have found the speed of a car is 300 \ mph you have most likely made an error as this answer is not ‘sensible’ within the context of the question as it is too high speed for a car. This is a useful (and quick) way of checking your answer.

### Practice formula for speedquestions

1. Find the speed of an object which travelled 240 \ km in 3 \ hours. Give your answer in km/h

720 \ km/h 80 \ km/h 0.0125 \ km/h 80000 \ m/hour \begin{aligned} \text{Speed } &= \text{ distance } \div \text{ time} \\\\ \text { Speed }&=240 \mathrm{~km} \div 3 \text { hours } \\\\ \text { Speed }&=80 \mathrm{~km} / \mathrm{h} \end{aligned}

2. Find the speed of an object which travelled 500 \ m in 5 \ hours. Give your answer in m/h

100 \ m/h 2500 \ m/h 0.01 \ m/h 0.1 \ km/h \begin{aligned} \text{Speed } &= \text{ distance } \div \text{ time} \\\\ \text { Speed }&=500 \ \mathrm{m} \div 5 \text { hours } \\\\ \text { Speed }&=100 \ \mathrm{m} / \mathrm{h} \end{aligned}

3. Find the speed of an object which travelled 1200 \ m in 3 \ hours. Give your answer in km/h

400 \ m/h 2500 \ m/h 0.4 \ km/h 2.5 \ km/h 1200 \ m=1.2 \ km

\text{Speed } = \text{ distance } \div \text{ time}

\begin{aligned} \text { Speed }&=1.2 \ \mathrm{km} \div 3 \text { hours } \\\\ \text { Speed }&=0.4 \ \mathrm{km} / \mathrm{h} \end{aligned}

4. A person walks at a constant speed. They travel 900 \ m in 0.5 \ hours. Calculate the speed. Give your answer in metres per minute (m/min)

30 \ m/h 1.8 \ m/min 1800 \ m/min 30 \ m/min \begin{aligned} 0.5 \text { hours }&=30 \text { minutes } \\\\ \text { Speed }&=\text { distance } \div \text { time } \\\\ \text { Speed }&=900 \ m \div 30 \text { minutes } \\\\ \text { Speed }&=30 \ \mathrm{m} / \mathrm{min} \end{aligned}

5. A person walks at a constant speed of 3 miles per hour. They walk for 90 \ minutes. Calculate how far they walk.

270 \ miles 4.5 \ miles 9 \ miles 45 \ miles 90 \ minutes = 1.5 \ hours

\begin{aligned} \text {Distance }&=\text {speed} \times \text {time} \\\\ \text {Distance }&=3 \times 1.5 \\\\ \text {Distance }&=4.5 \ \text{miles} \end{aligned}

6. An object travels 1800 \ m at a speed of 6m/s. Calculate the time taken. Give you anwer in minutes.

300 \ minutes 50 \ minutes 3 \ minutes 5 \ minutes \begin{aligned} \text {Time}&=\text {distance} \div \text {speed} \\\\ \text {Time}&=1800 \ \mathrm{m} \div 6 \text { m/s } \\\\ \text {Time}&=300 \ \mathrm{sec} \end{aligned}

300 \ seconds   is  5 \ minutes

### Formula for speedGCSE questions

1. Emily drives her car 186 \ kilometres in 3 \ hours. What is her average speed?

(2 marks)

186 \div 3

(1)

62

(1)

2. Jenny left her home at 9am and walked to the library. She arrived at 10:30am. The library is 3 \ miles from her house.

(a) What was Jenny’s average speed?

(b) On the return journey she takes a longer route home which is 2 \ miles longer than before. She takes 2 \ hours to walk home.

On which journey was Jenny walking faster?

(5 marks)

(a)

1.5

(1)

3 \div 1.5

(1)

2

(1)

(b)

2.5 \ mph

(1)

Faster walking to the library

(1)

3. Seobin was driving to a hotel.

He looked at his Sat Nav at 12:30, it said he had 66 \ km left for his journey.

Seobin arrived at the hotel at 15:48

Work out his average speed between 12:30 and 15:48

(4 marks)

Attempt to find the time taken for the journey

(1)

3 \ hours \ 18 \ minutes   or  3.3 \ hours

(1)

66 \div “3.3”

(1)

20

(1)

## Learning checklist

You have now learned how to:

• Use compound units such as speed
• Solve simple kinematic problem involving distance and speed
• Change freely between related standard units (eg time, length) and  compound units (e.g. speed) in numerical contexts
• Work with compound units in a numerical context

## The next lessons are

• Calculating density
• Calculating pressure
• Working with speed, distance, time questions

## Still stuck?

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#### FREE GCSE maths practice papers (Edexcel, AQA & OCR)

8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.

Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.