Graphical Analysis of Motion
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Familiar Mathematical Relationships
Less than
2 < 9
Two is less than nine
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Familiar Mathematical Relationships
Equals
g 
2
9.81m/s
Gravity on earth
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Familiar Mathematical Relationships
 Greater than
 Not equal to
 Less than or equal
 Greater than or equal
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NEW Mathematical Relationship

Proportional to
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Three Types of Proportions
yx
direct proportion
1
y
x
indirect proportion
(inverse)
yx
2
direct proportion to a
square
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Graphical Analysis of Motion
Now slowly,
one at a time…
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The Direct Proportion
“y is directly proportional to x”
yx
y x
y x
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A Direct Proportion Example
In straight line motion,
d t
“distance is directly
proportional to time”
t d
Given more time, you can drive farther
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Changing a Proportion into an Equality
Once again, consider…
yx
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Changing a Proportion into an Equality
yy 
x
 kx
1. Replace  with “=”
2. Multiply by a constant “k”
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Changing a Proportion into an Equality
y  kx
The constant “k” is an arbitrary letter…
essentially, we could have used the symbol of
our choice, “m” for example…
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Changing a Proportion into an Equality
y  mx
Look familiar? It should.
The “m” indicates the slope of a straight line,
which is exactly what a graph would look like
if plotted.
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Direct Proportion’s Graph
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Recognizing a Direct Proportion - 1
When the variables in question are …
On opposite sides of the equal sign
and
both in numerator or both in denominator
m  D V
mass  volume
Vf  a  t
Vf  t
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Recognizing a Direct Proportion - 2
When the variables in question are …
On the same side of the equal sign
and
one in numerator and one in denominator
m
D
V
d
V
t
mass  volume
distance  time
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Take a Breather
Get ready for the next
proportion.
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The Indirect (inverse) Proportion
“y is indirectly proportional to x”
1
y
x
y x
y x
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An Indirect Proportion Example
In accelerated motion,
1
a
t
a t
Given less time, your acceleration rises
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Changing a Proportion into an Equality
k1

y
xx
1. Replace  with “=”
2. Multiply by a constant “k”
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Changing a Proportion into an Equality
k
y
x
Y
X
1
1
0.5
2
0.33
3
0.25
4
0.20
5
0.167
6
For argument sake, let “k” =1
Look at how the values of “y” vary
What do you think this plot will look like?
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Indirect Proportion’s Graph
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Indirect Proportion’s Graph
hyperbola
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Recognizing an Indirect Proportion - 1
When the variables in question are …
On opposite sides of the equal sign
and
one in numerator and the other is in
denominator
Δv
a
t
1
a
t
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Recognizing an Indirect Proportion - 2
When the variables in question are …
On the same side of the equal sign
and
both in numerator or both in denominator
V  a  t
1
a
t
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Take a Breather
Get ready for the next
proportion.
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The Direct Proportion to a Square
“y is directly proportional to
x-squared”
yx
y x
2
y x
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The Direct Proportion to a Square Example
In accelerated motion,
d t
d t
2
As time increases, the distance traveled
also increases
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Changing a Proportion into an Equality
 kx
y
x
22
1. Replace  with “=”
2. Multiply by a constant “k”
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Changing a Proportion into an Equality
yx
2
Y
X
1
1
4
2
9
3
16
4
25
5
36
6
For argument sake, let “k” =1
Look at how the values of “y” vary
What do you think this plot will look like?
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Direct Proportion to a Square Graph
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Direct Proportion to a Square Graph
parabola
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Recognizing a Direct Proportion to a Square - 1
When the variables in question are …
On opposite sides of the equal sign
and
both in numerator or both in denominator
1 2
d  at
2
distance  time squared
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Recognizing a Direct Proportion to a Square - 2
When the variables in question are …
On the same side of the equal sign
and
one in numerator and one in denominator
2
v
ac 
r
radius  velocity squared
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Proportion Practice
Identify any and all proportions from the
following physics equations.
F  ma
E  mc
2
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Proportion Practice
Identify any and all proportions from the
following physics equations.
mv
Fc 
r
2
m4 r
Fc 
2
T
2
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Proportion Practice
Identify any and all proportions from the
following physics equations.
m1m2
Fg  G 2
d
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Graphical Analysis of Motion Practice
Qualitative Graphical Analysis
Back to Smart Notebook
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