I AM STILL THE WALRUS
Another time-elapsed photograph of a busy street at nighttime. The photo illustrates the velocity of vehicles as they travel the distance in the photograph over an unknown period of time.
I am the Walrus...
Goo Goo Ga'Joob
Ga'Goo Goo Ga'Joob
Ga'Goo Goo Ga'Joob.
("I Am The Walrus," by J. Lennon/P.McCartney, 1967)
This PTOA Segment encourages PTOA Readers and Students to conquer any fear related to definitions defined by mathematical expressions.
Using the familiar concept of Velocity, PTOA Readers and Students will learn that mathematical expressions are actually helpful with regard to defining common concepts used in ordinary daily life.
MATHEMATICAL RELATIONSHIPS AND SYMBOLS
Somewhere around the age PTOA Readers and Students attended middle school, the math classes related to "adding" and "subtracting" and "regrouping" and "factoring" and "decimals" and "fractions" concluded and the real fun began.
The course called Algebra I challenged students to determine how much Unknown Stuff, Unknown Length/Distance, or Unknown Time would result given a list of Things That Were Known about the situation.
To make the process less cumbersome than writing words like "Stuff" or "Length" or "Time" over and over, symbols were used for shorthand...
like "X" and "Y"...
and "A" and "B"...
and " α " (Greek letter called "Alpha")
and " β " (Greek letter called "Beta").
These symbols were inserted into mathematical expressions to solve for the Unknown.
These symbols were also called Variables, because their magnitude could "vary" depending upon their relationship with the Things That Were Known.
Unfortunately at that point math started to seem too abstract and became an exercise in suffering.
After that point, any mathematical expression that involved alphabet letters or symbols just caused great anxiety.
Only a coincidence that this woman with math anxiety has blond hair. Math concepts can confuse anybody. Your understanding of math is directly related to how well your math teachers understood the applications of math in the real world.
Too bad the instructor didn't clarify that "X" and "Y" and "A" and "B" and " α " and " β " were just shorthand for "an Unknown amount of Stuff or Length or Time."
And double ditto too bad that the instructor never explained that the mathematical relationship written on the chalk board/smart board was just an expression of how humans think the Universe works.
The above picture shows how math is used to model how a baseball will fly through the air.
The math used to predict the baseball's trajectory was postulated by a dude named Galileo in the late 1500s.
However, a stone batted with a club by a Cave Boy or Cave Girl followed the same trajectory.
In summary:
The above picture is also an example of how the rules of the Universe have always been around and humans developed mathematical relationships to better understand how Stuff (aka Mass and Matter) moves over a Length or Distance during an interval of Time.
Alphabetic and/or Greek symbols may be used in the mathematical expressions because they are easy for humans to move around and manipulate while trying to figure out how the rules of the Universe work.
For Example...
NEED FOR SPEED
PTOA Readers and Students are so familiar with the concept of "Speed" that they do not even think of it as a defined mathematical relationship.
Yet a definition of "Speed" was found to be helpful regarding expressing a commonly understood rate of distance travelled over a given timeframe.
"Speed" also has the slightly fancier name "Velocity."
Oh no! Not again!!!
When the policeman pulled you over for speeding, s/he clocked your car moving at a constant 85 miles per hour in a 55 miles per hour speed zone.
(Gee, you are in so much trouble!)
Miles is a unit of Distance.
Hour is a unit of Time.
Here's the mathematical relationship humans developed to define travelling at a constant Velocity (aka Speed):
Velocity (aka Speed) = Distance/Time
or in mathematical speak:
V = d/t
where V = Velocity d= Distance t=Time
To recap:
Policeperson states your V = 85 miles/hr.
You argue that your V = 55 miles/hr.
One thing is certain: You have a court date!
SCARCELY ODD (MEANING NOT ODD AT ALL)
PTOA Readers and Students will continue the PTOA Heat Transfer Focus Study Area.
The definitions of Heat Transfer via Conduction, Convection, and Radiation will require the use of mathematical expressions that define these concepts ... just exactly like the concept of "Velocity" is defined by a relationship between Distance and Time.
There is no need to be intimidated by the definitions and mathematical expressions.
Did you notice that Velocity is also a "Rate" because Time is in the denominator? Yes indeed, "Velocity" does inform the rate at which Stuff is moving.
The symbols used in the expressions will always be defined just like the d=Distance and t=Time defined the V=Velocity.
The tried and true method for decoding a mathematical definition is to first try to understand how the symbols work together to define the given entity:
In the above graphic, Velocity was defined as "a Distance" divided by "the Time" in which the distance was travelled.
Next PTOA Readers and Students should imagine what would happen to the defined concept if the magnitude of the individual symbols grew larger or smaller ...
If:
the d=Distance has a bigger number than it did before and the t=Time interval stays the same,
Then:
V = Velocity will be a larger number.
Hey! That means that the Velocity is faster than it was before!
On the other hand...If:
The t=Time in the denominator becomes a larger number and
The d=Distance travelled stays the same,
Then:
V is Velocity will be a smaller number.
Hey! That means that the Velocity is slower than it was before!
Wow! Both of these observations seem to accurately describe what happens in the real world regarding travelling at a constant Velocity.
That means that the math model that describes the definition of "Velocity" seems to accurately explain what happens in the real world!
Always remember that the mathematical expressions used to define concepts were developed by human beings in their continual attempt to decode the rules of the Universe.
TAKE HOME MESSAGES:
Mathematical expressions are used to define commonly known concepts and some not-so-commonly known concepts.
For example, below expression defines V=Velocity (aka Speed) as a d=Distance divided by t=Time:
V = d/T
The upcoming PTOA Heat Transfer Focus Study Area will likewise include definitions for Heat Transfer via Conduction, Convection,and Radiation using mathematical expressions.
PTOA Readers and Students can easily understand the mathematical expressions used in the upcoming Heat Transfer definitions by employing the same analytical techniques that were used in this PTOA Segment to rationalize how "Velocity" seemed to be defined accurately.
©2015 PTOA Segment 00061
Process Industry Applied Math
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