The Problem with
Acceleration
James Vesenka (jvesenka@une.edu)
Department of Chemistry and Physics
University of New England
Physics
Biddeford, ME 04005
NQLB – Orono 6/23-24/11
1
Motivation
 UNE: Modeling Instruction
 Student centered, guided inquiry
 Studio physics: observe & analyze 1st
 Emphasis on Multiple Representations
 A continuing challenge: The Ratio
 Velocity, acceleration, etc. (units)
 Slope of graph (units)
 Trigonometry & Scaling (comparison)
 Assessment: TUG&K
NQLB – Orono 6/23-24/11
Page 2
Naïve Physics Student
gravity
torque
F=ma
Disconnected
Factons
kinematics
impulse
energy
projectiles
centripetal
force
vectors
momentum
rotation
units
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Experienced Physicist
“Expert”
Thinker
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
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Model Approach: Mechanics
Conservation of Energy
ΔE = W + Q + R
Constant Force Particle
v≠constant ΣF=ma
Restoring Force
Particle: F=-kΔx
Free Particle
Impulsive Force
v=constant ΣF=0 Particle: ΣFΔt=Δp
Central Force
Particle: F=mv2/r (in)
NQLB – Orono 6/23-24/11
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E.g. Tumble Buggy
 Reality <-> Perception <-> Mental Model
Verbal
Physical
Phenomena
Diagrammatic
Graphical
Mental
Picture
Mathematical
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
1.
2.
3.

1.
2.
3.

Tumble Buggy Motion
What do you observe?
Moves at constant speed
Moves (mostly) in a line
Lights blink, wheels turn, noisy…
What can you measure (units too)?
Position (“X” in meters)
Time (“t” in seconds)
Buggy color, blink frequency, etc…
What are the constants? (speed)
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How does __ depend on __?
  So we have “variables” X and t.
  Which variable is independent (you
control). Which variable is dependent
upon the variable you control?
  Depends on how you collect data…
  We will use a metronome as our
timer:
http://www.metronomeonline.com/
NQLB – Orono 6/23-24/11
Page 8
Whiteboard Discourse
Diagrammatic
3
+X(m)
2
Graphical
y = mx + b
X=(+0.4m/s)t+0.5m
2
1
t=0
Δx
1
<-b
Mathematical
Δt
x = vt + xi
<v>≡Δx/Δt
Average
t(s) velocity
0
2
4
Verbal: For every second that passes the buggy
travels +0.4 meter every second from an initial
position of 0.5 meter in the positive x direction.
NQLB – Orono 6/23-24/11
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Summary of
Modeling Cycle
Consensus
Pre-lab
Paradigm Lab
Operational
Whiteboard
Discourse
Application
Graph, Math
Exam
Definitions
Diagram, Verbal
Refine
No
Test-Works?
NQLB – Orono 6/23-24/11
Yes
Page 10
Interactive and Phun
constructivist
vs
transmissionist
cooperative inquiry
vs
lecture/demonstration
student-centered
vs
teacher-centered
active engagement
vs
passive reception
student activity
vs
teacher demonstration
student articulation
vs
teacher presentation
lab-based
vs
textbook-based
Satisfies state and national learning results
Materials and Assessments: modeling.asu.edu
Modeling Workshop Kennebunk Maine:
http://www.uukennebunk.org
NQLB – Orono 6/23-24/11
Page 11
The Problem of the Ratio
 What Arnold Arons has to say.
 As applied to velocity and accleration.
 How to help visualize kinematics.
 The Motion Map.
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Arnold Arons
 “One of the most severe and widely
prevalent gaps in cognitive development
of students at secondary and early
college levels is the failure to have
mastered reasoning involving ratios.”
 “This disability, among the very large
number of students who suffer from it,
is one of the most serious impediments
to their study of science.”
Teaching Introductory Physics – Chapter 1 Section 6
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Verbal Representation
 Arons: “Linguistic elements play an
essential, underlying role in the
development of the capacity for
arithmetical reasoning with ratios and
proportion.”
 Lawson3 (1984): “a necessary …
condition for the acquisition of
proportional reasoning during
adolescence is the prior internalization
of key linguistic elements or
argumentation.”
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Arons Ctd.
Failure to do so  Fear of Mathematics
 “This problem will not be rectified until
we, in the colleges and universities,
produce elementary teachers who have
mastered arithmetical reasoning
sufficiently thoroughly to lead their
pupils into articulating lines of reasoning
and explanation in their own words.”
 Solution: Use Whiteboards dammit!
Teaching Introductory Physics – Chapter 1 Section 10
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Whiteboards
 Can examine different dependencies
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Clicker ?
 A ball is thrown upward (+z) and returns to
earth. The motion described below is AFTER
the ball leaves the hand to BEFORE it is
captured by the hand. What is the direction
of acceleration at three parts of its motion?
Top
Up
Down
Motion of
Ball
1
2
3
4
5
Up
down
down
down
up
up
Top of
Flight
down
0
0
up
0
Down
down
down
up
up
down
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Old Lab
 Dynamics track and cart, photogates or
motion detector.
 Beautiful data and graphs…
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Old Lab
 …little understanding.
 …math models can be problematic
Slope
Value
Math Model
x vs. t2 (1/4)a
0.092 m/s2
YUCK!
v vs. t
a
0.24 m/s/s
v = at
v2 vs. x
2a
0.45m2/s2/m
v2=2a∆x
Motivation: Annenburg CPB’s Mechanical Universe
Online Metronome
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New Lab
 Low Tech Solution.
 Beautiful graphs and data…
 …great math & diagrammatic models
using a highly visual and tactile process.
Slope
Value
x vs. t2 (1/2)a
0.024 m/s2
v vs. t
a
0.046 m/s/s
v = at
v2 vs. x
2a
0.089m2/s2/m
€
v2=2a∆x
NQLB – Orono 6/23-24/11
Math Model
Δx =
1
2
at2
Page 20
Dangerous Short Cuts
 An object travels a distance d in an arbitrary time
interval t to define the average velocity and
acceleration:
v =
d
t
→a =
v
t
→d =
1
2
at 2
 “The students are not informed that the meaning of
the symbols was changed in the derivations that
followed, and many emerge with little understanding
of either the physical concepts of velocity and
€
acceleration or the algebraic equations.”
 It is much safer to be explicit:
< v >=
Δx
Δt
→a =
Δv
Δt
→ Δx =
NQLB – Orono 6/23-24/11
1
2
aΔt
2
Page 21
Faster and Faster Particle
4
+x
16 x(m)
3
How do we
linearize this
data?
Workshop
9 tomorrow
2
4
1
t=0
1
0
1
NQLB – Orono 6/23-24/11
2
3
4
t(s)
Page 22
“Kinematics Stack”
Bonus:
a≡
=
=
=
=
Δv
4
+x x(m)
16
Δt
vf − vi
9
tf − ti
−
3
4
1
8 v(m/s)
t(s)
0
t(s)
3s − 2s
+
3s − 2s
2
1
a v t=0
a(m/s/s)
2
0
NQLB – Orono 6/23-24/11
t(s)
4Page 23
Motion Map the Following
+x x(m)
80
60
40
20
0 v(m/s)
t(s)
0
t(s)
a(m/s/s)
0
1
NQLB – Orono 6/23-24/11
2
3
4
t(s)
Page 24
t=0 a v
a≡
=
=
=
=
Δt
vf − vi
Motion Map Result
1
x(m)
80
2
60
Bonus:
Δv
+x
tf − ti
40
3
−
20
t(s)
t(s)
0
0
3s − 2s
+
3s − 2s
4
-40 v(m/s)
0
-10
1
a(m/s/s)
NQLB – Orono 6/23-24/11
2
3
4 t(s)
Page 25
“Kinematics Stack”
Bonus:
a≡
=
=
=
=
Δv
Δt
vf − vi
tf − ti
−
3s − 2s
4
3
2
1
+
3s − 2s
t=0
a v
+x x(m)
16
9
4
1
8 v(m/s)
0
-2
a(m/s/s)
NQLB – Orono 6/23-24/11
t(s)
t(s)
t(s)
4Page 26
TUG&K Results (R Beichner)
 Test for Understanding Graphs in Kinematics:
 less (technology)
 is more (understanding).
9;):)6*."
9-6*:)6*."
748)"2345+."
<*=>5-"?2@$'A"
BCD)E*=;)"?2@F!A"
/)001-"20-66."
:;4>5,-+40"?2@&GA"
/)001-"2345+."
()*)+,-+."
!"
#!"
$!"
%!"
NQLB – Orono 6/23-24/11
&!"
'!!"
'#!"
Page 27
David Dellwo: https://www.iupui.edu/~josotl/index.php
NQLB – Orono 6/23-24/11
Page 28
Acknowledgements
 Thanks to Danielle Parent, Aubrie
Dickinson, Ashley Ruggieara, Kerra
Gearinger & Shawna Hatfield
 Thanks to Susan McKay and Maine
RISE Center for the invitation.
 Thank you for listening.
NQLB – Orono 6/23-24/11
Page 29
Faster and Faster Workshop
x(m)
Observables:
Prediction
How does _____ depend on _____
Procedure:
NQLB – Orono 6/23-24/11
0
t(s)
- v(m/s) +
Constants:
t(s)
- a(m/s/s) +
Measureables:
t(s)
Page 30
Faster and Faster Workshop
Data Table:
x(m)
0
Results
Linearized
t(s)
Math Model:
0
- v(m/s) +
Math Model:
t(s)
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v(m/s)
Faster and Faster Workshop
Linearized
Math Model:
∆x(m)
0
Motion Map:
Results Grid:
+x
0
Consensus Math Models:
slope
x vs t2
v vs t
v2 vs ∆x
NQLB – Orono 6/23-24/11
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