MEI Conference 2016
Preparing to teach motion
graphs
Sharon Tripconey
[email protected]
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Preparing to teach motion graphs
June 2016 © MEI
Kinematics content extracts for reformed Mathematics A level from Sept 2017
Extracts taken from ‘Mathematics AS and A level content’ (2014), DfE.
www.gov.uk/government/publications/gce-as-and-a-level-mathematics
Note that bold text within [square brackets] is AS content.
P Quantities and units in mechanics
Ref
P1
Content description
[Understand and use fundamental quantities and units in the S.I. system: length, time,
mass]
[Understand and use derived quantities and units: velocity, acceleration, force, weight],
moment
Q: Kinematics
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
[Use calculus in kinematics for motion in a straight line: 𝒗 =
𝒂=
Q5
𝒅𝒗
𝒅𝒕
=
𝒅𝟐 𝒓
,
𝒅𝒕𝟐
𝒅𝒓
,
𝒅𝒕
𝒓 = ∫ 𝒗 𝒅𝒕, 𝒗 = ∫ 𝒂 𝒅𝒕 ]; extend to 2 dimensions using vectors
Model motion under gravity in a vertical plane using vectors; projectiles
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Preparing to teach motion graphs
June 2016 © MEI
displacement
0
velocity
time
0
time
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Preparing to teach motion graphs
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Preparing to teach motion graphs
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Some useful links:
Moving man:
https://phet.colorado.edu/en/simulation/moving-man
Traffic graphs (GeoGebra):
http://www.geogebra.org/m/2832353
MEI’s M4 magazine, Issue 49:
http://mei.org.uk/files/pdf/Sept-Oct-2015-1.pdf
Integral resources **NEW** Walkthroughs:
http://integralmaths.org/walkthroughs/index.php?wt=vel_time_graphs
http://integralmaths.org/walkthroughs/?wt=vuat
http://integralmaths.org/walkthroughs/?wt=disp_const_acc
http://integralmaths.org/walkthroughs/index.php?wt=disp_time_graphs
Information and professional development:
http://www.mei.org.uk/2017-pd
http://www.furthermaths.org.uk/teaching-mechanics
http://www.mei.org.uk/files/pdf/mechanics-get-set-course-outline.pdf
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Preparing to teach motion graphs
June 2016 © MEI
Preparing to teach motion
graphs
Sharon Tripconey
Session description
This session is one of four designed to help teachers prepare for teaching
mechanics topics in the new A level from 2017.
‘Preparing to teach motion graphs’ will cover some of the basic subject
content, links to GCSE and other A level topics, as well as exploring ideas
and approaches for teaching this topic. The session will demonstrate how
simple practical classroom activities can provide a stimulus for students to
develop their understanding of motion graphs.
This session is particularly suitable for teachers who have not previously
taught any mechanics.
position
distance travelled
Describing quantities
Scalar quantity
distance
distance travelled
speed
time
Vector quantity
displacement
position
velocity
acceleration
It is not a specific requirement within GCSE Mathematics to know the
terms ‘vector’ and ‘scalar’ quantities and understand the distinction, but
it is in GCSE Science.
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.”
Mathematics GCSE subject content (DfE)
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.”
Reformed A level Content
Mechanics content
Q: Kinematics
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]
[Understand, use and derive the formulae for constant acceleration for
motion in a straight line]; extend to 2 dimensions using vectors
Q3
Q4
Q5
Model motion under gravity in a vertical plane using vectors; projectiles
www.gov.uk/government/publications/gce-as-and-a-level-mathematics
Displacement, distance &
distance travelled
displacement
0
time
Displacement, distance &
distance travelled
distance
displacement
distance travelled
0
time
What is
significant
about the
gradient in
these
graphs?
Displacement & position
position
displacement
0
time
Language of motion
Average speed
=
Average velocity =
total distance travelled
total time taken
total displacement
total time taken
Displacement & distance travelled
displacement
distance travelled
Average speed = total distance travelled
total time taken
0
time
Average velocity = total displacement
total time taken
Velocity and speed
velocity
0
time
Velocity and speed
velocity
speed
0
time
Comments
• A distance–time graph has less information than a
displacement–time graph.
• You cannot deduce displacement from distance travelled.
• Similarly, you cannot deduce velocity from speed.
• It is easy to confuse graphs of
velocity-time
(v – t)
and
displacement-time
(y – t)
with
displacement-displacement
(y – x)
Extending basic ideas
The instantaneous velocity is the gradient of the
displacement – time graph
displacement s m
5
0
5
time t s
Acceleration
Acceleration is a measure of how much velocity is changing.
This means it can affect both the speed and direction of
motion.
If we only consider 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.
Exercise
This is a velocity-time graph for the journey of an object
moving in a straight line.
Describe the motion as fully as you can.
Constant velocity =
4m/s (gradient is
zero)
Area=displacement
= 10+12+8 = 30m
gradient = acceleration
+0.8ms-2
v=0 indicates a
change in
direction
Area=displacement
= -(4.5+9.5) = -14m
gradient = acceleration
-1 ms-2
Area 9.5m
(approx)
Moving Man
https://phet.colorado.edu/en/simulation/moving-man
Acceleration
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.
Key features of velocity-time
graphs
• GRADIENT represents acceleration
• AREA UNDER GRAPH represents displacement
Summary
Motion
Graph
Displacement
-time
Gradient
Velocity
Velocity-time
Acceleration
Accelerationtime
Rate of change of
acceleration
Area
Notes
Vertical axis can
Not significant be positive or
negative
Areas below the
time axis
represent negative
Displacement displacement. v=0
indicates a
possible change
in direction
Velocity
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