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Math 334 Lecture #1
§
1.1: Mathematical Modeling; Direction Fields
The Steps of Mathematical Modeling.
Consider an object falling in the atmo
sphere near sea level.
[NOTE the assumptions: object falling; in the atmosphere; near sea level.]
Step 1
: Identify the dependent and independent variables.
Let
v
be the velocity of the falling object. [This describes the “falling” of the object.]
Let
t
be the time.
As the velocity presumably changes with time, velocity is a function of time:
v
=
v
(
t
)
.
Step 2
: Choose consistent units of measurement for the variables.
Measure
t
in seconds (sec or s).
Measure
v
in metres per second (m/s).
Step 3
: Articulate the physical law or principle that governs the physical process.
The motion of the falling object is governed by Newton’s second law:
sum of the forces =
ma,
where
m
is the mass of the object, and
a
is its acceleration.
[This introduces a “parameter”
m
and the variable
a
.]
Step 4
: Express the physical principle in terms of the variables declared in Step 1.
Since acceleration is the time derivative of velocity, it follows that
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
 Spring '11
 Smith
 Math

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