Mechanics:

Motion graphs

Simon Clay

simon.clay@mei.org.uk

Session description

In 2017 the new A level in Mathematics will have a compulsory

mechanics component.

In this session participants will explore some of the prior learning

of students at GCSE related to motion graphs. We will also

consider some of the misconceptions students hold and address

issues associated with the specific language and terminology of

motion graphs.

This session is one of four sessions designed for teachers who

have not previously taught mechanics with the aim of helping to

prepare them for teaching mechanics topics in the new A level.

The content of this session is adapted from the one-day Get Set:

Mechanics course which has run this year.

distance travelled

position

Ref Content description

Q1 [Understand and use the language of kinematics: position; displacement;

distance travelled; velocity; speed; acceleration]

Q2 [Understand, use and interpret graphs in kinematics for motion in a straight

line: displacement against time and interpretation of gradient; velocity

against time and interpretation of gradient and area under the graph]

Q3 [Understand, use and derive the formulae for constant acceleration for

motion in a straight line]; extend to 2 dimensions using vectors

Q4

Q5 Model motion under gravity in a vertical plane using vectors; projectiles

Q: Kinematics

Reformed A level Content

www.gov.uk/government/publications/gce-as-and-a-level-mathematics

Basic ideas and language of motion

The distance is the physical distance between the chosen

points.

Displacement is the shortest route between two points.

Position describes the location of something relative to a

fixed point. This point is usually referred to as the origin.

The distance travelled between two points does not have to

be the same as the distance between two points, described

above.

time0

displacement

Displacement-time graph

Task: Now add distance & distance travelled graphs

time

displacement

distance travelled

0

distance

Displacement, distance & distance

travelled

The Moving Man

Displacement, distance & distance

travelled

time

displacement

distance travelled

0

What is

significant

about the

gradient in

these

graphs?

distance

Mathematics GCSE subject content (DfE)

14

Plot and interpret graphs (including reciprocal graphs and

exponential graphs) and graphs of non-standard functions

in real contexts to find approximate solutions to problems

such as simple kinematic problems involving distance,

speed and acceleration

“Awarding organisations may use any flexibility to increase depth,

breadth or context within the specified topics or to consolidate teaching

of the subject content.”

15.

Calculate or estimate gradients of graphs and areas

under graphs (including quadratic and other non-linear

graphs), and interpret results in cases such as

distance-time graphs, velocity- time graphs and

graphs in financial contexts

“Awarding organisations may use any flexibility to increase depth,

breadth or context within the specified topics or to consolidate teaching

of the subject content.”

Mathematics GCSE subject content (DfE)

Average speed = total distance travelled

total time taken

Average velocity = total displacement

total time taken

Basic ideas and language of motion

Displacement & distance travelled

time

displacement

distance travelled

0

Average velocity = total displacement

total time taken

Average speed = total distance travelled

total time taken

Velocity and speed

time

velocity

0

Velocity and speed

time

velocity

speed

0

Extending basic ideas

• The instantaneous velocity is the gradient of the

displacement – time graph

time 0

displacement

t s

s m

Acceleration is a measure of how much velocity is changing.

This means it can affect both the speed and direction of

motion.

When considering motion along a straight line only two

directions are possible, either forwards or backwards.

An acceleration of 2 ms-2 means that the velocity of a particle

increases by 2 ms-1 every second (by 2 metres per second per

second).

For example, if a car has an initial velocity of 6 ms-1 and an

acceleration of 2 ms-2, then after 1 second its velocity will be 8

ms-1, after 2 seconds 10 ms-1 and after 3 seconds 12 ms-1 etc.

Acceleration

Describe the motion as fully as you can.

ExerciseThis is a velocity-time graph for the journey of an object

moving in a straight line.

What the significance of

these sections of the

graph being straight

lines?

What can we say about this

stage of the motion when

the graph is curved?

Velocity

If a particle has a negative acceleration but a positive

velocity, then it will slow down to a stop and then move in

the opposite direction, with its speed steadily increasing.

Take care with the word deceleration. It is probably better

not to use it. Use negative accelerations instead!

Take care that the units of acceleration are ms-2. This is

usually read as ‘metres per second squared’, or sometimes

as ‘metres per second per second’.

Accelerations can be found using the gradients of

velocity-time graphs.

Acceleration

Key features of velocity-time graphs

• GRADIENT represents acceleration

• AREA UNDER GRAPH represents displacement

gradient = acceleration

+0.8ms-2

gradient = acceleration

-1 ms-2

Area=displacement

= -(4.5+9.5) = -14m

Constant velocity =

4m/s (gradient is

zero)Area=displacement

= 10+12+8 = 30mv=0 indicates a

change in

direction

Area 9.5m

(approx)

SummaryMotion

GraphGradient Area Notes

Displacement

-timeVelocity Not significant

Vertical axis can

be positive or

negative

Velocity-time Acceleration Displacement

Areas below the

time axis

represent negative

displacement. v=0

indicates a

possible change

in direction

Acceleration-

time

Rate of change of

accelerationVelocity

Time to reflect….

About MEI

• Registered charity committed to improving

mathematics education

• Independent UK curriculum development body

• We offer continuing professional development

courses, provide specialist tuition for students

and work with employers to enhance

mathematical skills in the workplace

• We also pioneer the development of innovative

teaching and learning resources

MEI Conference 2017

Mechanics: Motion graphs

Simon Clay simon.clay@mei.org.uk

2 of 7 SC 08/06/17

Final version

Session description: Motion graphs

In 2017 the new A level in Mathematics will have a compulsory mechanics component.

In this session participants will explore some of the prior learning of students at GCSE related to motion

graphs. We will also consider some of the misconceptions students hold and address issues associated

with the specific language and terminology of motion graphs.

This session is one of four sessions designed for teachers who have not previously taught mechanics

with the aim of helping to prepare them for teaching mechanics topics in the new A level.

The content of this session is adapted from the one-day Get Set: Mechanics course which has run this

year.

Useful links

‘Mechanics in Action’ https://www.stem.org.uk/elibrary/resource/26065

This is a fantastic resource that is available electronically from the National STEM Centre. It is freely

available but you do have to set up an account or log in to be able to access it. It discusses modelling in

mechanics and gives a number of practical experiments that can be done with students. It includes

photocopiable handouts and details the theory behind the experiments as well as working through

solutions and results.

PhET Interactive Simulations https://phet.colorado.edu/

This website contains a series of interactive simulations which are useful for a number of different

mechanics topics. They are simple to use and free.

One example is ‘The Moving Man’ which is useful for illustrating motion graphs:

https://phet.colorado.edu/en/simulation/legacy/moving-man

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Session reflections

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Final version

Some basic motion definitions

The distance is the physical distance between the

chosen points. It is a scalar quantity, requiring no

specified direction.

Displacement is the shortest route between two

points. A distance and direction are needed and so it

is a vector quantity.

Position describes the location of something relative

to a fixed point. This point is usually referred to as

the origin. A distance and direction are needed and

so it is also a vector quantity.

The distance travelled between two points does not have to be the same as the distance between two

points, described above. If the route is not direct, then the distance travelled will be greater than the

direct distance between the two points. Distance travelled is a scalar quantity.

Speed is a measure of how the distance travelled is changing with respect to time. It is a scalar

quantity. Speeds are always positive with just the size given and no direction given or implied.

The velocity is the rate at which the position changes. Velocities are vector quantities with a magnitude

and direction and can be either positive or negative.

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 = 𝑇𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑

𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑇𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛

Acceleration is the rate at which the velocity changes. This means it can affect both the speed and

direction of motion. Note that an acceleration of 2 ms-2 means that the velocity of a particle increases by

2 ms-1 every second (by 2 metres per second per second).

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Displacement, distance & distance travelled graphs

0

displacement

time

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Velocity & speed graphs

time 0

velocity

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A velocity-time graph exercise

This is a velocity-time graph for an object moving in a straight line. Describe the motion as fully as you can.