skubis © 2003 graphical analysis of motion. skubis © 2003
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Graphical Analysis of Motion
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< 92Two is less than nine
Familiar Mathematical Relationships
Less than
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9.81m/s2gGravity on earth
Familiar Mathematical Relationships
Equals
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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
direct proportionxy
xy
1 indirect proportion
(inverse)
2xy direct proportion to a square
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Now slowly,
one at a time…
Graphical Analysis of Motion
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The Direct Proportion
“y is directly proportional to x”
xy xy
xy
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A Direct Proportion Example
In straight line motion,
td “distance is directly
proportional to time”
dt Given more time, you can drive farther
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Changing a Proportion into an Equality
Once again, consider…
xy
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Changing a Proportion into an Equality
xy kxy 1. Replace with “=”2. Multiply by a constant “k”
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Changing a Proportion into an Equality
kxy The constant “k” is an arbitrary letter…essentially, we could have used the symbol ofour choice, “m” for example…
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Changing a Proportion into an Equality
mxy 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 signand
both in numerator or both in denominator
tVt aV
volumemassV Dm
ff
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Recognizing a Direct Proportion - 2
When the variables in question are …
On the same side of the equal signand
one in numerator and one in denominator
timedistance t
dV
volumemass V
mD
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Take a Breather
Get ready for the next proportion.
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The Indirect (inverse) Proportion
xy
1
“y is indirectly proportional to x”
xy xy
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ta
1
ta
An Indirect Proportion Example
In accelerated motion,
Given less time, your acceleration rises
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Changing a Proportion into an Equality
xy
1x
ky
1. Replace with “=”2. Multiply by a constant “k”
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Changing a Proportion into an Equality
x
ky
For argument sake, let “k” =1Look at how the values of “y” vary
Y X
1 1
0.5 2
0.33 3
0.25 4
0.20 5
0.167 6
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 signand
one in numerator and the other is in denominator
t
1a
t
Δva
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Recognizing an Indirect Proportion - 2
When the variables in question are …
On the same side of the equal signand
both in numerator or both in denominator
t
1at aV
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Take a Breather
Get ready for the next proportion.
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“y is directly proportional to x-squared”
xy xy
The Direct Proportion to a Square
2xy
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2td
td
In accelerated motion,
As time increases, the distance traveled also increases
The Direct Proportion to a Square Example
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Changing a Proportion into an Equality
2xy 2kxy
1. Replace with “=”2. Multiply by a constant “k”
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Changing a Proportion into an Equality
2xy
For argument sake, let “k” =1Look at how the values of “y” vary
Y X
1 1
4 2
9 3
16 4
25 5
36 6
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 signand
both in numerator or both in denominator
squared timedistance at2
1d 2
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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 signand
one in numerator and one in denominator
squared velocityradius r
va
2
c
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Proportion Practice
Identify any and all proportions from the following physics equations.
maF 2mcE
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Proportion Practice
Identify any and all proportions from the following physics equations.
2
24
T
rmFc
r
mvFc
2
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Proportion Practice
Identify any and all proportions from the following physics equations.
221
d
mmGFg
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Graphical Analysis of Motion Practice
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