skubis © 2003 graphical analysis of motion. skubis © 2003

SKUBIS © 2003

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


Equals

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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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xy kxy 1. Replace with “=”2. Multiply by a constant “k”

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kxy The constant “k” is an arbitrary letter…essentially, we could have used the symbol ofour choice, “m” for example…

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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


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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xy

1x

ky

1. Replace with “=”2. Multiply by a constant “k”

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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



one in numerator and the other is in denominator

t

1a

t

Δva

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Recognizing an Indirect Proportion - 2




t

1at aV

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Take a Breather

Get ready for the next proportion.

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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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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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Recognizing a Direct Proportion to a Square - 2



one in numerator and one in denominator

squared velocityradius r

va

2

c

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