Truth. Volume II (Proper)

8/7/2019 Truth. Volume II (Proper)

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ROUGH DRAFT

A Seeker's Truth

Sudhir Gajanan Murthy


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Table Of Contents

0 Theoretical Introduction to Truth

1 Theoretical Introduction to Formal Observation

2 Theoretical Introduction to Classical Kinematics

3 Theoretical Introduction to Classical Forces

4 Theoretical Introduction to Classical Energy

5 Theoretical Introduction to Classical Momentum

6 Theoretical Introduction to Classical Rotations

7 Theoretical Introduction to Fluid-Dynamics

8 Theoretical Introduction to Thermo-Dynamics

9 Theoretical Introduction to Classical Waves

10Theoretical Introduction to Classical Electrostatics

11 Theoretical Introduction to Classical Circuits

12 Theoretical Introduction to Magnetostatics

13 Theoretical Introduction to Special Relativity

14 Theoretical Introduction to Electro-Magnetism

15 Theoretical Introduction to General Relativity

16 Theoretical Introduction to Nuclear Physics

17 Theoretical Introduction to Quantum Mechanics

18 Theoretical Introduction to Cosmology

19 Theoretical Introduction to Matters and Compositions

20 Theoretical Introduction to Ambiguities with Truth Casualties in Occurence


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0

Theoretical Introduction to Truth

The Seeker: Father, now read me that book, for I can understand its language

God: Yes my Son, now that the language has been understood, its craft can begin

God: This book is called The Universe: Once upon a time, there was


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Curiosity. Allows high moments and moment's high by letting weed even get a try. Allows us to question, to

imagine which without I would not passion. Allows stories to be told in many ways, even if Truth itself has only one

say. Curiosity. The drive that drives us all. The wings whose never fall. The meaning that stands infinitely tall.

Curiosity.

When Man first came upon the world, or when World first came upon the man, this curiosity was all that would

demand. And were the first seekers of Truth created, to solve Truth's puzzle by which we were painted. By so began

the inevitable quest, whose mandate to seek the Truth was God's test. Piece by piece Truth's empty puzzle lay flat,

unknown is next had us never relaxed. But to the first humans gravity not be known, and even the common still be

sewn. And until unwoven stays, the fabric of knowledge awaiting later days.

Soon the corner pieces of Truth be fit, which led to the realization that you can build from it. So the puzzle began to

deepen, as man's thirst for Truth began to steepen. Finally, the puzzle began to make sense, but the unused piecesbegan to gather dense. As with every answer you get two questions, with every piece fit in Truth's puzzle you get

two more pieces of tension. But at that time, the picture of the puzzle could be predicted, enough so that rules are

rules of restricted. For now, you can create the puzzle piece and see if it fit, instead of having to find the piece out of

all the infinite.

Dreamers, who have long been awakened, still consider this method, called deduction, forsaken. Like math without

numbers or English without letters, deduction works as simple as breads with butters. Though flawed as kinematics

without velocity or fluid dynamics without viscosity, deduction cannot be interpreted as logic without curiosity.

Without Curiosity. Allows high moments but moment's low by letting weed even get a go. Allows us not to

question, nor to imagine which with I cannot passion. Allows stories to be told in only a way, even if Truth itself has

many says. Without Curiosity. The drive that kills us all. The wings whose only fall. The meaning that stands

infinitely small. Without Curiosity.

I would be deeply enraged, if I see this is where we have swayed.

Have you ever seen a Hero so righteous, that he cannot do any more justice to the world because he already made

the world so perfect, there is no more good to be done. Or have you ever seen a Villain so corrupted, that he is

incorruptible because there is nothing more to corrupt him with. But then again, what is the true difference

between a Hero and a Villain? What is the true difference between right and wrong? What is the true difference

between black and white but a shade? A shade that we define. A shade that we put limits and bound by an

everlasting sense of ethics. A shade that creates a line of morality, that distinguishes right from wrong. Right

cannot exist without wrong. For a Hero cannot be a Hero, unless there is a Villain. But sometimes, a Hero loses his

sense of morality, as he realizes meaningless in existence. Sometimes, a Hero forgets what he is fighting for, as he

discovers a hidden realm underneath all others. Sometimes, a Hero looks beyond the darkness in night, as he

becomes a seeker of truth. Sometimes, a Hero must become a White Knight.

For science and religion both questions the same question. Both odes the same ode. Both seeks the same seek.Both truths the same truth. For I see one, but functions as none, whose difference, between science and religion , are the separate answers to the same question: truth. But truth is not knowledge nor laws nor equations- but


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understanding of nature and the formulation of its derivations. For there are reasons for things to be. An entity topurpose you see. Often called God but truth to me.

(During school in a class, the matter of beliefs concerning truth raised


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Sudhir: I wonder too, for I do not know.

Teacher: See Sudhir, that is of which is God. For science can only answer so many questions but they approach the

one unknown- God.

Sudhir: Just because something is unknown does not necessarily mean that it was created by God. And for that

matter let us define God to be this unexamined, that is the unknown. Then it follows that truth manifests in God asGod manifests in truth. So this truth is also this unknown- this unmanifest. Then "intelligence comes into being

when the brain discovers its fallibility, when it discovers, what it is capable of, and what it is not." (-Krishnamurti).

Hence this is all God's manifestation of Truth. Truth's manifestation of Truth. Truth. That is, the unknown. Then

truth is the unknown as God is the unexamined, so God and truth are of but equal concepts. Under such parameter,

I ode not just to truth, I ode to God. I seek not just to truth, I seek to God. I believe not just in truth, I believe in God .

"whose surface is Truth, but within is God"

"It is easy to complicate the simple, but difficult to simplify the complex"

"Instead of asking 'What is Truth?', it is easier to ask 'What is not Truth?'"

"Did God create Man or did Man create God?"

"The source of great evil is the source of greater good."

If mathematics is language, then physics is craft. But if someone were to ask me which I prefer more, I wouldsimply reply we learn how to write in order to describe; we learn how to read in order to comprehend; so I learn

how to math in order to physics.

It takes one man to speak the truth, but it takes half the population to be the truth.

Truth is not my cause, it is my effect.

If God were a light bulb, they would be its Thomas Edison. If God were physics, we would be the laymen. If Godwere true, I would be false.

Is lack of knowledge innocence, or is fulfillment of knowledge corruption? Then I would rather teach the manwith the curious heart, then the man with the arrogant mind. Then I would rather myself to observe passionately,

then to think ignorantly. Then I would rather heart, then brain; for the former can create the latter, while the lattercan only destroy the former.

There is nothing more grand, nothing more noble, nothing more tolerable, than nothing more but to seek truth.

To live a life that is not a seeker of truth, is to live a life that is not worth living.

With the sun pointed opposite, and the stairs never infinite, even shadows collapse. For even shadows distort as

they climb, in a discontinuous manifestation, step by step. Yet at the stairs which lead no stairs, and shadows

merge below into the hollow of inner peace, all was but an interruption- like a subtle ripple in a momentous sea.


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When all is little, littled and weak

it is the truth that we must seek.

When all is riddle, riddled the shown

it is the truth that must be known.

When all is fiddle, fiddled is lore

it is the truth that is our core.

With eyes I shall always seek the truth. With mouth I shall always speak the truth.

With ears I shall always hear the truth. And with heart I shall never fear the truth.

But even if eyes be blinded, mouth be torned, ears be deafed, and heart be numbed,

With god I shall always be the truth.

I understand that anything else but to seek the truth is tautological and so one must detach from the meaningless manifestations within truth,yet without loss of the wholeness of truth. But I find myself in a situation I had thought already be completed, and that is to detach thyself from.I have detached myself from most , and most emotions to only its remains of compulsions, passions, and sorrows. But there has been an emotionthat blinds the seeker of truth, wrought by childhood and simulated by loneliness, hidden in its very foundations- the overwhelming, irrevocableand unconditional love. I speak not of love as the passion, but love as an object attracted to thy loneliness, and I must detach myself from love astolerably as I have the other emotions and manifestations. To whom this love is aimed at is fixed- or perhaps the image of a shadow- and withcrafting hesitation of fears and shedding sorrows of tears, I must detach myself from whom I love, as to be a seeker of truth. I understand that all logic and emotions matter not, but that the truth is of only meaning and that is which I seek with all sincerity and genuine affection, but Ihave been told by the closeness of my honesty that passion fades. Regardless of this meaningless love to an individual, or attachment to themysticism and spirituality of things- I must, even with love or hate, continue to seek the truth. If my passion ever begins to sway from truth, I will still with all sincerity and genuine affection, continue to be a seeker of truth. For the lack of all else, and the presence of just one greed- to seek the truth- is deemed not by its magnitude, but by its existence; so even a small lotus is but the only one large lotus untouched by water. If I ever

become detached from love in all contexts, I will still with all s incerity and genuine affection, continue to be a seeker of truth. For I do not wish to

exist in states of attachment with only acknowledgments of the necessity and purity of detachment; as would be living in truth even in an awareenlightenment, without seeking the truth in full enlightenment. If I be blinded on the preference to seek happiness over the truth, I will still withall sincerity and genuine affection, continue to be a seeker of truth. For I cannot allow myself to interfere with thyself; as the source of great evil is the source of greater good. If by any means I surrender to these meaningless manifestations, I will still with all sincerity and genuine affection,continue to be a seeker of truth. For I cannot adhere within truth, but I must adhere without- in order to be its seeker; as I cannot surrender theresults to god, I must surrender god to the results. If by any bias or prejudice I come to pass these trials of god, I will still with all sincerity and genuine affection, continue to be a seeker of truth. For it is not that I must run away, it is that I must run to; as meaningless as the lesser truth, Imust be its seeker to and for the greater truth. We are not the hands of gods, nor the feats of demons, but the hearts of men. And with every heart in this man-I will still with all sincerity and genuine affection, continue to be a seeker of truth. To be cleansed as a man that is living, isbetter than to be cleansed as a man that is dying. For I understand my words here, allows there to exist all so that is good and yet all that is evil.

I will still with all sincerity and genuine affection, continue to be a seeker of truth: this is my Oath and my Ode to the truth of all plural truths.

I come to see, the truth with me

but all I find, is that Im blind.

There is a difference between a cripple that is whole and a whole that is cripple: the former is what the latter

seeks.

PS: I do not believe in god just use god to personify truth. (though im open minded to enlightment and truth)


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The entire universe, in order to be understood, has to be expressible. In order to be expressible, it has tobe logical. In order to be logical, the universe must be simplified. Yet the most understandable,expressible, and logical method of simplifying concepts of the observed, is mathematics. Yet mathrequires quantities, variables, and numbers- so then the question arises: how can the universe bequantified? There are three fundamental, observable quantities to the universe: Length, Time, and Mass

of which scientists can measure and therefore quantify. These are actually a quality to a quantity,otherwise known as dimensions which are incompatible. For you cannot compare the taste of food tothe smell of flowers- they are of different qualities. Likewise, length cannot be compared to time normass and vice versa.

[ ] [ ] [ ] For how can a certain length represent a certain timeor a mass equating to a certain length? Theycannot because they are values independent to each other. Just like smells cannot abstractly equaltastes because they are different genres or fields, these physical quantities cannot equal other becauseof their independent quality, that is, they have different dimensions. So a dimension is defined to be aphysical nature of a quantity; in the observable universe, all quantities constitute from thesefundamental dimensions: Length, Mass, and Time.

A unit of length represents distance- this quantity can be expressed visually as length is the distancebetween two points (let us say A and B). Scientists call this measure of length with units such as feet,meters, miles, etc.; all those unit quantities represent the dimension of length, that is they measuredistance.

That length represents, specifically, how much is between object A and object B.

A unit of mass represents how much is within a(n) object(s) or the amount of "manifestation" or"occupancy" within a given space- (let us say A and B). This amount of how much is in there is calledmatter, so mass is (for now, which implies that later we will find another definition of mass) themeasure of how much matter is within a substance. The specific units for mass are grams, kilograms,etc; all of which represents the dimension of substance within an object

Notice that the mass of object B is greater than the mass of object A because there is more within Bthen there is A.


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(Note: it should not necessarily mean that size dictates mass as shown in the figure above; considerrocks stacked in a box- no matter the size of the box the mass depends the amount of rocks if weassume the box has no initial mass. In this sense, size does not directly relate to mass in a necessarymanner)

A unit of time represents how long an interval has under gone. The rate at which time changes is aconstant(for now we can say that time is a constant, however in the latter aspects we shall return to itsown relativity) and quantifies how long events are (let us say A and B). The specific units for time areseconds, hours, etc; these all measure the interval of events.

The time interval itself is to express how much "time" has elapsed, in other words how long an eventoccurred.

However, notice that there are multiple units for the same dimension such as length. There might bemeters, inches, feet, miles, furlongs, leagues, fathoms, nautical miles, etc. Yet, since these are all units of length and a measurement of distance, the question arises what is the international units. For example,U.S.A's currency is in dollars while Indian currency is in Rupees and these currencies functionindependently; however, math and physics is an international and universal concept- what should bethe universal units? With length, physicists want to deal with units that can be applied universally formajority of observable phenomena, and eventually decided the standard international unit for lengthshould be in meters . Since meters are not too big or too small (in our daily activities), they can quantifylong and small distances easier than, say, miles. Though miles can be used to express long distancetravel, it would not be effective or practical using its units to find the distance between a person's feet

to that person's head, and since practical physics involves neither that of the too big nor the too smallbut of just the relative "average", the length must represent such. Similarly, scientists agreed that fortime, the universal units be seconds, and for mass, the universal units be kilograms. These standard andinternational units themselves are called the SI Units (International System of Units). Below is a tableshowing SI Units for the three basic dimensions: length, mass, and time.


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However, the scopes of SI base units also are limited to the average size. For instance, it would be a

terribly difficult number to express in base SI units the distance between the Earth to the Sun. Also, itwould be a terribly small number to express the distance between two atoms in base units. So then,scientists decided to put prefixes onto the SI units that either multiply or divide it.

meter m length l

kilogram kg mass m

second s time t

Standard prefixes for the SI units of measure

Multiples

Name deca- hecto- kilo- mega- giga- tera- peta- exa- zetta- yotta-

Symbol da h k M G T P E Z Y

Factor 10 0 10 1 102 103 10 6 109 10 12 10 15 1018 10 21 1024

Subdivisions Name deci- centi- milli- micro- nano- pico- femto- atto- zepto- yocto-

http://en.wikipedia.org/wiki/SI

http://en.wikipedia.org/wiki/SI
http://en.wikipedia.org/wiki/Metrehttp://en.wikipedia.org/wiki/Metrehttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/SI_prefix#List_of_SI_prefixeshttp://en.wikipedia.org/wiki/SI_prefix#List_of_SI_prefixeshttp://en.wikipedia.org/wiki/Deca-http://en.wikipedia.org/wiki/Deca-http://en.wikipedia.org/wiki/Hecto-http://en.wikipedia.org/wiki/Hecto-http://en.wikipedia.org/wiki/Kilo-http://en.wikipedia.org/wiki/Kilo-http://en.wikipedia.org/wiki/Mega-http://en.wikipedia.org/wiki/Mega-http://en.wikipedia.org/wiki/Giga-http://en.wikipedia.org/wiki/Giga-http://en.wikipedia.org/wiki/Tera-http://en.wikipedia.org/wiki/Tera-http://en.wikipedia.org/wiki/Peta-http://en.wikipedia.org/wiki/Peta-http://en.wikipedia.org/wiki/Exa-http://en.wikipedia.org/wiki/Exa-http://en.wikipedia.org/wiki/Zetta-http://en.wikipedia.org/wiki/Zetta-http://en.wikipedia.org/wiki/Yotta-http://en.wikipedia.org/wiki/Yotta-http://en.wikipedia.org/wiki/Deci-http://en.wikipedia.org/wiki/Deci-http://en.wikipedia.org/wiki/Centi-http://en.wikipedia.org/wiki/Centi-http://en.wikipedia.org/wiki/Milli-http://en.wikipedia.org/wiki/Milli-http://en.wikipedia.org/wiki/Micro-http://en.wikipedia.org/wiki/Micro-http://en.wikipedia.org/wiki/Nano-http://en.wikipedia.org/wiki/Nano-http://en.wikipedia.org/wiki/Pico-http://en.wikipedia.org/wiki/Pico-http://en.wikipedia.org/wiki/Femto-http://en.wikipedia.org/wiki/Femto-http://en.wikipedia.org/wiki/Atto-http://en.wikipedia.org/wiki/Atto-http://en.wikipedia.org/wiki/Zepto-http://en.wikipedia.org/wiki/Zepto-http://en.wikipedia.org/wiki/Yocto-http://en.wikipedia.org/wiki/Yocto-http://en.wikipedia.org/wiki/Yocto-http://en.wikipedia.org/wiki/Zepto-http://en.wikipedia.org/wiki/Atto-http://en.wikipedia.org/wiki/Femto-http://en.wikipedia.org/wiki/Pico-http://en.wikipedia.org/wiki/Nano-http://en.wikipedia.org/wiki/Micro-http://en.wikipedia.org/wiki/Milli-http://en.wikipedia.org/wiki/Centi-http://en.wikipedia.org/wiki/Deci-http://en.wikipedia.org/wiki/Yotta-http://en.wikipedia.org/wiki/Zetta-http://en.wikipedia.org/wiki/Exa-http://en.wikipedia.org/wiki/Peta-http://en.wikipedia.org/wiki/Tera-http://en.wikipedia.org/wiki/Giga-http://en.wikipedia.org/wiki/Mega-http://en.wikipedia.org/wiki/Kilo-http://en.wikipedia.org/wiki/Hecto-http://en.wikipedia.org/wiki/Deca-http://en.wikipedia.org/wiki/SI_prefix#List_of_SI_prefixeshttp://en.wikipedia.org/wiki/Timehttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Lengthhttp://en.wikipedia.org/wiki/Metre


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Simply add the prefix before the unit of measure to either multiply or divide the unit base value. Forexample, kilometer means 1000 meters (or 10 meters) and millimeter means .001 meters (or 10`meters). With the standard prefixes for SI units, small or large quantities can be expressed practically.

However, what exactly can be defined a meter, a kilogram, or a second? For any arbitrary definition of

such basic dimensions, the tests results (corresponding to such definitions) need to be consistent. Sothese unit base quantities were given values to certain phenomena in the universe that always observedto have remained constant. For instance, a simple definition of a meter could be the distance of threesteps anyone walks; though there are so many variables: a persons foot may vary, a persons leg may belonger, a person might travel farther with each step, etc- and these all might cause the person to walkmore or less of the distance each time. Since there are many variables in that testing procedure,scientists want to pinpoint a definition of a meter with as close accuracy as possible. If they had gonewith 1 meter is 3 steps anyone walks, each individual persons may constantly arrive with different valuesof lengths for a meter, and then a meter cannot be universally understood nor accepted. So a metermust be some distance that when tested for again, will lead to the same result (or as close as feasible).Then scientists thought hard and asked themselves what definition should we give a meter such if wetest for it again and again, the results will be consistent? Since light (a phenomena with a constant speedotherwise known as the speed of light) travels at a constant speed (in a vacuum), if scientists measurehow far light travels in a certain time frame, then that can define a meter precisely such if they testagain and again for a meter, that data will be consistent. So the meter is the length of the path travelledby light in vacuum during a time interval of 1 299 792 458 of a second. If scientists test for a meter again andagain with this as the definition of a meter, their test results will be extremely accurate and precisebecause of the constant nature of light. For example, let us hypothetically situate that a specific caralways and constantly travels at a certain speed (let us say 60 miles an hour). Then if we want todetermine what the exact distance of 60 miles is, all we do is measure the length of the distance thatspecial car travelled after one hour; since the car travels at a constant speed of 60 miles per hour, thenafter one hour it should have traveled 60 miles. So if we measure that distance the car travels in onehour, we can properly define that to be 60 miles and if we test for it again, the car will still travel 60miles per hour and we will still get the same length the car traveled after one hour. But how can weknow that the car is travelling at 60 miles per hour? Light travels at some speed but it travels at aconstant speed and so regardless of however fast its travelling, we just arbitrarily define a meter to bethe distance light travels in 1 299 792 458 of a second- so our definitions are actually quite arbitrary.

Symbol d c m n p f a z y

Factor 10 0 101 10 2 10 3 10 6 10 9 10 12 10 15 10 18 10 21 10 24


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Using this concept of consistency to define a kilogram, scientists approach a dilemma - what substancein the universe is a constant 1 kilogram? Since there exists no such constant (or that such a constant ishard to locate and therefore difficult to define), scientists must artificially define a kilogram. So scientistsexercised a rather ingenious method- grab a specific substance and define it to be a kilogram. Forexample, we can define a kilogram to be the mass of the monitor in my house. Then from that, I can

approximate the mass of all other objects in my house by comparing them to my definition of akilogram- my monitor. My lamp might have a mass twice as much as the monitor- so (due to mydefinition of a kilogram) my lamp is 2 kilograms. With this whole respect, scientists decided touniversally call a specific substance- a platinum iridium alloy- the exact 1 kilogram, and in turn all othermasses are to be compared to this exact "value" of a kilogram. This specific platinum iridium that hasmass of exactly 1 kilogram is called the International Prototype Kilogram (IPK) which defines thekilogram (this IPK now resides in a vault at the BIPM in Svres) .

To define the second, a consistent natural phenomenon must exist such that the interval of that naturalevent is a constant. I can define a second to be the time interval it takes for Sudhir to clap his hands; so

the interval it takes to clap my hands is defined to be a second. However notice that sometimes I cantake a longer time to clap my hands or a shorter time, so a second will always vary. Then I must findanother phenomenon where if I test for the time interval again and again, the passage of time remains aconstant- so that interval can be defined as a second. Observations have shown such a natural property,that when a specific atom at a specific state exists, it radiates at constant intervals. So a second isdefined to be the duration of 9,192,631,770 periods of the radiation of the cesium 133 atom (at acertain state: when the Kelvin Temperature is 0 and there is no movement of the atom at ground state).

These are the basic dimensions of the universe which can be used to express and quantify nature. Asstated before, these dimensions have no correlation to each other, as they are independent, and so

mathematically:[ ] [ ] [ ]

This essentially means that meters can never equal to seconds or to kilograms, and vice versa, as thesephysical qualities have different meanings: length measures distance, time measures duration, and massmeasures matter within. However, the dimensions length can equal other dimensions of length, masscan equal mass and time can equal time. This can simply be mathematically expressed:

[ ] = [ ] [ ] = [ ]

[ ] = [ ]

As 1 second equals 1 second, the same dimensions can equal the same dimensions. It would beincorrect to say 1 second equals 1 meter because they both measure different qualities. So dimensionsmust be analyzed to "check for equality". If a physicist declares that 300 seconds equals 4 meters, thenthrough this process of analyzing the dimensions, one can conclude this declaration is incorrect as timecannot equal length. This process is called dimensional analysis and can be used to check whether
http://en.wikipedia.org/wiki/Bureau_International_des_Poids_et_Mesureshttp://en.wikipedia.org/wiki/S%C3%A8vreshttp://en.wikipedia.org/wiki/Caesiumhttp://en.wikipedia.org/wiki/Caesiumhttp://en.wikipedia.org/wiki/S%C3%A8vreshttp://en.wikipedia.org/wiki/Bureau_International_des_Poids_et_Mesures


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dimensions of the answer are correct. If this physicist had declared that 300 seconds is 5 minutes, thenthrough dimensional analysis, one can conclude that he may be right because both dimensions measurethe same quality: time. However, if the physicist declares that 300 seconds is 300 minutes, throughdimensional analysis he STILL may be right, since both the dimensions measure the same quality: time.Even though 300 seconds does not equal 300 minutes, dimensional analysis does not involve the

numbers or quantities, but their units or qualities; that is, their dimension.

If a student suddenly tells his teacher that: 275 34.21 = 456 , the teacherwould show the student false, dimensionally, because 275 meters is a measurement of length, 34.21seconds is a measurement of time, and 456 kilograms is a measurement of mass. So by substituting thisback into the student's argument, we find that the student urges: [ ][ ] = [ ].However, this statement is false because [ ][ ] only equals a [ ][ ] , not amass. If another student suggested that 275 34.21 = 1110 234 ,then the teacher can show that dimensionally this statement can be correct as meters measures length,seconds measures time, feet measures length, and milliseconds measures time. So dimensionally, the

student is claiming: [

]

[ ] = [

]

[ ] which is a true statement. Through

dimensional analysis, this Student's units or dimensions are consistent, so his statement might becorrect. For dimensional analysis does not guarantee whether a formula is correct, but only shows thatit might be correct or is incorrect altogether. In another example, as student might claim 5 meters isequal to 6 meters; using dimensional analysis, this student might be correct since both these quantitiesmeasure length so both their dimensions are the same but notice how the student's argument is falsebecause clearly 5 meters does not equal 6 meters. Hence, dimensional analysis cannot confirmstatements or formulas, but only check for equality up to a point whether they might be correct orwhether they are incorrect. For the dimensions only refers to units, not the number itself anddimensional analysis only checks to see if the units are correct, not the actual numbers themselves.

Using these concepts of dimensional analysis and the basic dimensions themselves, scientists have beenable to conceive ingenious ideas using the three basic dimensions: Length, Time, and Mass. Bymanipulating concepts of length and time dimensions mathematically, scientists forged the beginningbut never ending path of physics called kinematics. Defined to be the mechanics of movement of anobject without reference to the causation of movement, Kinematics just deals with abstract concepts of how normal objects move, not about what caused the objects to move. Essentially, kinematics is thestudy of objects movement and has been cleverly derived by the concepts and manipulation of thefundamental dimensions.


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2

Theoretical Introduction to Classical Kinematics

God: Not moving is as inherent as moving.

The Seeker: How so?

God: The former is a special case of the latter.


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In order to understand kinematics, we must recompose it. For kinematics is just the answers to previousscientific questions, and since we must re-derive the answers, we must re-ask the questions thescientists have asked. For scientists have asked: How fast is an object travelling? What does it mean totravel fast? What does it mean to travel slow? These answers which scientists have forged have lead towhat is now the founded kinematics. Yet, since we must develop kinematics for our own understanding,

we must answer these same questions. How fast is an object travelling? What does it mean to travelfast? What does it mean to travel slow? If an object travels fast, that means that the object covers a lotof distance in a short amount of time. Likewise, if an object travels slow, that means that the objectcovers little distance in a long amount of time. Scientists coined the term for how fast an object istravelling, as speed. Mathematically and logically, the concept of speed can be defined as:

= where 's' refers to speed; 'd' refers to distance covered; and ' t' refers to the time elapsed. For

simplicity, let us just consider

as t because they both mean the time elapsed.

=

All that this expression means is characterized under the following: imagine two particles in emptyspace, objects 1 and 2, both of which are simply moving. Object 1, let us say, has more speed thanobject 2. If object 1 is travelling 10 meters per second and object 2 is travelling 5 meters per second,then after 1 second object one would have covered 10 meters and object 2 would have covered 5meters. Object one would have travelled a greater distance than object 2 in the same amount of timesimply because of their respective speeds. Generalizing the speeds to any arbitrary values- 1and 2forobjects 1 and 2, respectively, all speed measures is a distance covered per the time it takes to travel thatdistance. Hence speed is just some length over some time- the rate at which an object moves. With thecase of object 1 and object 2, for any arbitrary speeds they travel, the concept of speed is the same- themeasure of how much distance they covered per how much elapsed time it took. At some instance,denoted instant 1, Object 1 and Object 2 are recorded as follows-


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These objects at this instance lie on the shown coordinates, or simply they exist in that spatial order.Since these objects are, as stated, moving, let us see where the objects are located after some time haspassed (some arbitrary time denoted as t ). After this t amount of time, the object's positions arerecorded as follows-

At this latter instant (or after some amount of time), called instant 2, notice that both objects movedfrom the previous position after some time. Yet scientists inquire "how fast did these objects move?" Soa logical representation of this "how fast the objects went" or speed is distance over time. Essentiallyhow much length these objects are covering with respect to how long it took. As we can see, after someamount of time object 1 travelled farther than object 2 traveled in the same amount of time. Hence,Object 1 is travelling faster than object 2, in other words the speed of object 1 is greater than that of object 2. Let us expand this to pure logic or mathematics and combine Instant 1 with Instant 2 to have agreater understanding.

Let us denote that distance traveled by Object 1 as 1and the distance traveled by Object 2 2.Since thetime at which these positions were recorded is the same to both objects (i.e. the objects traveled duringthe same time interval), it is clear that Object 1 has traveled a greater distance than Object 2,so

1> 2


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If I divide both sides by t (the time it took the object to move these respective distances), then we canget a better logic of speed

1> 2 For notice how distance divided by time is speed. The distance object one traveled divided by the time ittook to travel that distance is simply the speed of object 1; likewise, the distance object 2 traveleddivided by the time it took to travel that distance is simply the speed of object 2. So this equality aboveleads to this logical statement:

1> 2 Or the speed of object 1 is greater than the speed of object 2. Though it might seem obvious that thespeed of object 1 is greater than object 2 (as one can clearly see in the pictures), this mathematicalrepresentation of speed should then become obvious. For we can dimensionally just rewrite speed as

=[

][ ]

that is, speed just measures how much length is travelled in a certain amount of time. So with thisbetter understanding of speed (how much time an object takes travel a certain length- how fast anobject is going), then we can mathematically manipulate it to form another concept. Since speed is

=

If we multiply t to both sides of this equation we can form this concept:

( ) = ( )

=

So if an object is travelling at some speed for some amount of time, the distance that object traveled isits speed by its time. So if an object were traveling 1meter per second, than the distance the object

covered after one second is simply 1 meter. If the object travels at the same speed for, now, twoseconds, than the object travels a distance of 2 meters. Using this concept of distance and its relation to

speed with respect to time has allowed scientists to formulate another concept. Let this thoughtexperiment be as follows: take an object traveling ,in empty space, at some arbitrary speed s for some

arbitrary time t . Then the distance d this object travels is simply d = st . However, the question is asked-in what direction does this object travel? For instance, speed measures distance to time, yet distancemeasures just a length. I can travel 1 meter walking forward but I can also travel one meter walkingbackward and I can travel 1 meter in any direction. Yet that direction is ignored because all speed isconcerned is with the distance, not the direction of the distance. For let us say I 1 travel at a speed 1

meter per second for 1 second. Then the distance I traveled is 1 meter, but that 1 meter can be in anydirection. For speed does not tell us the direction of where an object is traveling because speed is based


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off distance and distance is not based on direction, but that of length. If I walk north for 1 meter in thetime interval of 1 second, then my speed is 1 meter per second. And if I walk south for 1 meter in 1

second, my speed is also 1 meter per second, and if I walk east or west or any direction for 1 meter in 1second, my speed is always 1 meter per second. My speed does not change based off the direction I

travel in due to this relationship between distance and direction: distance does not measure direction.

So this object can be travelling that speed in any direction: North, South, East, and West or any directionin between. Though its speed can be in any direction, as its distance traveled can also be in any

direction, the distance itself is a constant since = .

Basically, this object can be travelling at the speed s in any direction since speed does not quantifydirection. So the actual distance covered would be as this:

The object could travel that distance, d (or d=st ), in any direction. So the object traveling at some speedfor some amount of time obviously covers some amount of distance. But since speed only measures thedistance an object travels to the time it took to travel that distance, speed cannot measure direction. Soscientists realized that speed just measures how fast something travels and not what direction it travelsin. Hence, speed is known as a scalar quantity because it represents a certain magnitude without anydirection itself. Yet scientists want to measure the speed as well as the direction speed is in. They want


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to convert speed into a vector quantity, a number that gives both magnitude and direction, in this case avalue that gives both speed and direction of the speed. With this, comes the ideal that developedvelocity.

Physicists want the velocity to measure both speed and the direction of speed, yet mathematically how

can this concept be expressed? We know velocity cannot equate to distance over time as that just tellsus speed and not the direction, so what formula for velocity can express not only speed but also thedirection of speed? The brilliance of mankind has articulated such an elegant mathematical definition:

= Where the represents average velocity, refers to change in position, and refers to change in

time or time elapsed. For simplicity, let us just consider as t since they both mean the time elapsed.

= This is very similar to the definition of speed, except that distance d is replaced with change in position,also called displacement . So in order to understand the difference between speed and velocity, wemust understand this difference between distance and displacement. Let us take this example todistinctly clarify between displacement and distance: take an object on a level surface-

Let us say that the object moves to the left at some speed for some amount of time. Then that meansthat this object will travel towards the left.

The distance between the object before and after it moved can be thought as taking a string to putbetween the object and measure the length of the string. Distance is simply the length between thesetwo points, in this case the two objects. Now try to think of an instance where distance is a negativevalue and eventually you will run into a conclusion that distance is never negative. For when in this


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universe can length between two points be negative? It can be zero, and some positive distance, butnever a negative distance between two objects. So this distance traveled by this object is a positivenumber. To show this, let us arbitrarily say that this distance between the objects lessened-

Notice that there is still some distance between the object before and after moving. Now let us evenlessen this distance:

The object still travels some positive distance, though a very minimal gap between the objects. Now letus say that there is 0 distance between the object and the object after travelling, in other words, let ussay that the object has not even moved.

Notice how the objects overlap, simply because the object has 0 speed so the object cannot travel anydistance. Now let us take the case where the object travels in the other direction

If the object travels in this direction now, then that means the object moves in the direction


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The key point here is that the distance the object travels is still positive. There is still length between theobject before and after it moved which means that the distance is forever and always positive (or zero if it did not move). So even if the object traveled farther outward, the distance is still positive becauselength between two points will and always will be positive (unless the length is 0)

So even if the object travels in this direction, the distance is positive. This means that distance is either 0(there is no length between) or positive. Notice how distance never reveals anything about direction asit is a positive magnitude. Since distance is used to measure speed, and distance cannot measuredirection, then speed can also measure direction. So the speed must always be positive or zero andcannot reveal anything about direction.

But displacement tells us the change in position which we can mathematically define. Take a NumberLine:

where the number line represents the position of the object. Since the object starts from 0, its initial

position is 0. Now displacement, remember, is change in position which logically means the finalposition minus initial position as this tells us how much the position has changed. Then it follows changein position is simply:

= or change in position is the final position minus the initial . In this case, the object on thenumber line has the initial position is 0. Now notice how the object has speed directed toward the right,


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and on the number line that would mean that for whatever final position, it too must be positivebecause the object is moving to the right (where we define on the number line to be positive). Since thechange in position is the final position minus initial position.

=

and final position is a positive number and the initial position is 0,

= [+ ] 0 = [+ ]

so velocity, which is change in position over the time it takes to travel, is:

=

=

[+]

However, notice that time t is a positive number because there is some interval between these eventsand time cannot be negative (time does not run backwards) so then velocity can simply be:

=[+][+]

and a positive number divided by a positive number is a positive number, so velocity is

= [+]

If, however, the object speeds in the opposite direction on this number line

then

= We know that the initial position of the object is 0, but the final position of the object or where theobject is after it moves is negative. Notice that the object will be travelling towards the left hand side of


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the number line and that position there is defined to be negative. So the final position is negative andthe initial position is 0. Substituting this in mathematically to the definition of displacement:

=

= [ ] 0

= [ ] In other words, the change in position is negative. Since velocity is

= and is a negative number in this case, we form

=[ ]

However, as discussed before, the time elapsed by the object is always positive

=[ ][+]

Since a negative number divided by a positive number is negative, velocity in this case is

= [ ] Basically the accounts for the direction, in other words, displacement includes not only the distance,but also the direction of the distance. So this negative velocity simply means that the object is travellingin the negative direction, or in this case the object is moving towards the left hand side. When thevelocity was positive, that simply meant that the object was travelling towards the positive direction, orin this case, the object moved towards the right hand side. So if I travel 1 meter forward, the distance Itraveled is one meter and the displacement is also one meter. If I, though, travel one meter backward,then the distance I traveled is one meter but the displacement is negative one meter. So the magnitudeor scalar of displacement is distance and distance with a direction, or the vector quantity of distance isdisplacement. Due to this, since speed is just distance over time and velocity is just displacement overtime, speed is the magnitude of velocity and speed with a direction is velocity itself. So a generalunderstanding of velocity is simply speed with a direction.

For example, on a freeway there is let us say a car that travels to the left covering 2 meters for every 1second. Then if I define everything to the left to be negative (like as the number line), the velocity of thecar is negative 2 meters per second. But notice how the speed of the car is just 2 meters per second asspeed is velocity without direction. Likewise, if the car is travelling to the right covering 2 meters every 1second, then the velocity of the car is positive 2 meters per second. And notice how the speed is also 2meters per second as it is velocity without direction.


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For now, let us consider motion of an object in just one dimension- as in the number line scenarios. Letthe object just move in one axes (up/down or left/right) and not on multiple axis since then, we asscientists will find too complicated (or perhaps not later on). However, let us master motion in just one-dimension to combine 2 one-dimensional motions together to form two dimensional motions. So let usproceed with some general inquiries regarding displacement. Notice how in the prior examples, the

velocity was strictly based off from displacement, and displacement was strictly based off somecoordinate system, (in that case, the displacement was based from the number line) but it comes to ourattention as to whether or not that coordinate system is an arbitrary decision. In other words, can we asan observer place any imaginary coordinate system on the space where the object moves? Would itaffect the velocity we measure? This question can be answered through a story as follows:

Take a crow that sits at some tree all by him/her self. The crow, however, wants to go to his other crowfriends but they are at another tree. Now there is a student's house in the middle between these twotrees. The student stands right outside his/her house and notices that the tree with the crow all alone isat the left side, and that tree with many crows is on the right hand side.

The red dot is the lonely crow, the black dots are the other friendly crows, and the blue dot is the

student observing these trees and crows. Then this student (who takes a physics course) coincidentallyhas a measuring rod and an accurate clock. Soon, the lonely crow moves from the tree to the other tree-and the student quickly measures the time the crow takes to travel from one tree to another (we willassume the student's data and measurements are accurate). The student finds that it took the crowexactly 1 second to travel to the other tree; then taking out his measuring rod, the student measures thedistance between the two trees to be 1 meter and then he concludes that the speed of the crow was 1meter per second. However, this student then decides to calculate the velocity, but since he must findvelocity, he must measure displacement. In order to measure a change in position, the student mustalready have a reference plane by which he/she can calculate the displacement. So this student decidesto create the reference frame as a number line where the position of the lonely crow is 0 so the final

position is 1 meter (since the distance traveled by the crow is 1 meter).


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Notice that the distance the crow travels is 1 meter, so even if the initial position is negative one, thefinal position would be just 0. So the displacement is

= = 0 ( 1) = 1

The displacement is the same 1 meter. So the velocity is the same 1 meter per second. Even if thephysics student changed the reference frame by which to measure the position of the crow, the changein position is the same. Even if the final and initial positions vary, the change in position is the same.Even if the student were too make up some now arbitrary reference frame, the change in position is still

the same. So the physics student tries just that- instead of creating some initial position as 0, 1, -1, orany specific number, the student decides to call it a variable.


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Notice how the final position, for any arbitrary position x_i, is simply x_i + 1. This is because the crow hastraveled 1 meter, so it follows that the final position must be one more than the initial position.

= + 1

=

= ( + 1) = 1

Since for any reference frame the displacement is still the same, the velocity also remains the same 1meter per second.

The moral of this story is simply no matter what reference frame, by which an observer measures theinitial and final positions, the change in position or displacement does not change. So velocity does notchange based off any random frame of reference by which the observer measures the object's position,

as the change in position will be the same in every case.

Now that this inquiry has been settled, let us extend the concepts of displacement and positions withrespect to velocity.

= =( )

= ( )

= +

This equation just means that the final position is simply the initial position plus the velocity by the time.If you notice, this statement basically implies how the final position varies based off initial position,velocity, and the time velocity was applied onto the object.

With this furthered understanding of the relationship between position to velocity, scientists decided tograph that data. This simple concept of velocity was further expressed by explaining position as arelation to time (position as a function of time). Basically, record and measure the object's position, astime changes and record and plot that data on a graph.

So scientists just observed the position of the object at certain points in time and graphically displayedthe observations as to embrace fuller control of the principles of velocity. For example, let us say anobserver records the position of the object at a specific time. Then the recorded position vs time graphwould graphically represent:


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Basically, there is an object that starts at some position (let us say 0). Then after some time that objectreaches another position. Then after some time a (some arbitrary time), the objects reaches a positionof x (an arbitrary position). In order to start from the position of 0 and reach a position x after some timea , the object must have moved, that is the object had to have had a speed and velocity. Now after timea , notice how the position of the object never changes. Since the position the object does not change,that means the object is not moving; that is, that object has a velocity of 0 (since there is 0 displacementor 0 change in position as the object does not change its position). Yet how can we figure out thisvelocity specifically? We are given the position and the time, so we must inquire into the logical andmechanical aspects of velocity- how is it that we can calculate this with such a graph? This graph issimply x(t) or basically that means how the position varies by time (position as a function of time) and

we know the definition of velocity:

= The relationship between this (velocity) and the graph lies within the slope. For the slope m of anyarbitrary function f(x) is

= ( ) but notice that f as a function of x ( ) in this case is simply x as a function of t x(t) and that is simply . So by substituting, the slope is

=


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So the slope of this graph (position as a function of time graph) is the change in position over the changein time. Notice this is the exact definition of an average velocity, so the slope of a position vs time graphis simply velocity:

=

= =

This simply means that if we were given a position vs time graph, its slope is the average velocity. Tofurther demonstrate, take another position vs time graph and using this concept of slope, let us find theaverage velocity from time to time where the object started at position and traveled to .

The slope of the position vs time graph is simply the change in position over the change in time which isthe exact definition of velocity. Measuring the slope between two points of the position vs time graph isof exact equivalence for measuring the average velocity between two points in time. So if in this case,the value of is 1 meter, the value of is 2 meters, the value of is 1 second, and the value of is 2seconds, then the average velocity this object is travelling is

=

=( )( )

=(2 1)(2 1)

=11


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This extended development of velocity to see how it relates to a position vs time graph led scientists tocoin what is known as instantaneous velocity. Let us assume a character named Alpha drives his car toschool one particular day. Also, let us record his position vs time graph while Alpha is driving.

First, Alpha hits the pedal and starts driving toward school, but then while driving; Alpha remembersthat he left his backpack home and so goes back home. After Alpha goes back home, Alpha then hits thepedal and goes on the freeway towards school. However, on the way Alpha encounters a stop sign andso stops for a long time and the resumes going towards school. Finally, Alpha makes it to school butsadly he is late and gets detention. Since we have recorded this data translated into a position vs timegraph, we can find the average velocity Alpha traveled throughout the whole trip.

This is the average velocity Alpha covered from starting at home, to reaching school, in other words thisis the average velocity of Alpha's trip. But what does this average velocity mean? Surely it is a change inposition traveled over the time it took to travel, but does that mean the average velocity is the actualvelocity. Surely there is some average velocity traveled here, but notice how the average velocity does


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not signify the actual velocity that Alpha travels. If, let us just arbitrarily assert, Alpha has an averagevelocity of 30 miles per hour, note that when Alpha was at the stop sign, he was not travelling 30 milesper hour but at that moment he was at the stop sign, Alpha was travelling 0 miles per hour. Then whenAlpha went on the freeway, Alpha was not travelling 30 miles per hour, but faster. When Alpha washeading back home because of the forgotten backpack, Alpha was not traveling positive 30 miles per

hour, but negative (as he was going backward). Notice how the average velocity does not reveal thevelocity Alpha was travelling at a moment, or at an instant. For even if Alpha took another path to get toschool, and the time it took to get to school and the change in position to get to the school is still thesame, then Alpha's average velocity is still the same.

So even if Alpha never had to go back home to get his backpack and there was no stop sign so hisdistance vs position graph looks like this, as long as the displacement to get to school and the timeinterval to get to school were the same, Alpha's average velocity does not change. So average velocitydoes not describe the actual velocity Alpha travelled to get to school, but rather just the "average" ormean of all the velocities that Alpha used to get to school.

If a person walks at a constant velocity of 1 meter per second for 1 second, and then the person travels3 meters per second for 1 second, then the person is displaced by 4 meters. Then the total change inposition is 4 meters and it took the person 2 seconds to change that much in position. So the averagevelocity is simply the change in position over the time it took to travel which is 4 meters over 2 secondswhich is 2 meters per second. The person had an average velocity of, therefore, 2 meters per second.But the person never traveled 2 meters per second, the person traveled as stated only 1 meter persecond and 3 meters per second

In another example, take two student's test scores on a physics test to be: 100 and 90. Then the averagescore is 95, but 95 does not mean it is the test scores of either of the students, it just tells the average ormean of total scores. So if a car travels a velocity of 100 miles per hour for one instant and then 90 milesper hour for the next instant, then average velocity of these two instances is simple 95 miles per hour.


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Though 95 miles per hour is not necessarily the velocity of the car at either instance, it is just theaverage or mean of those instances.

So now the question comes- how can one describe mathematically, the velocity at an instant? Asaverage velocity just describes the average rate at which position changes over a time interval,

instantaneous velocity describes the exact rate at which position changes for an instant- for a moment.But then how long of a time interval is an instant? For we know the definition and concept of velocitymathematically and logically:

= So if we know the time interval in which it takes the object to change a known position, than surelyvelocity can be measured. But since we want to find an instantaneous velocity or a velocity at an instant,what is the time interval for that instant? Over an instant or a moment, there is 0 change in time- in factthat is the very definition of an instant- where the time between two events is 0, that is there is no

interval. But over such a small interval, the object displaces very little as well. Since an instance is whenthere is no interval or no change in time, and there is no displacement, to calculate that velocity-

=00

(the symbol denotes velocity at an instant or instantaneous velocity)But even in mathematics this is an undefined number. Since there is mathematical ambiguity findingvelocity like this, where the time interval is 0 or when measuring velocity at an instant which makesdisplacement 0, we get an undefined answer. So then we must not use this direct algebraic approach as

to measuring velocity at an instant, but rather calculus. For though dividing by 0 is undefined, if we takethe limit as the time interval approaches 0, we can calculate a measurable instantaneous velocity.

= lim 0 =

(the symbol denotes velocity at an instant or instantaneous velocity)If we were given the displacement by time graph:


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Then the average velocity between the time interval [a,b] is simply the change in position between timeb and a over the time interval itself. But if we were to find the velocity at the instance when time is a,then simply let the time interval become very small. Since the average velocity is taking the slope of thisgraph between two points, if we want to find the velocity at a point, we must find the slope at thatpoint. But if we lessen the time interval such that it approaches 0, then we are taking the slope betweentwo points (in this case points a and b) but as those points approach each other (in this case as point bapproaches point a) it follows that taking the slopes between these points approaches the slope at onepoint (in this case, point a). To further demonstrate, let us lessen the time interval as to find theinstantaneous slope at time a:

Though this is a better "approximation" of what the velocity at might be, this is still an average velocitybetween two points. Since we want to find the velocity at time a, then let the other point approach


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point a and then take the slope between those two points. In other words, let the approach 0- so letus further lessen .

There is still some time interval , but it is relatively small so notice how this velocity, when the timeinterval lessens, approximates the actual velocity at a. If we were to take this one step further andsuggest that these points in time are so close together that there is 0 distance between them, that iswhen approaches 0, we can then take the slope between these two points. But since the points areactually just one single point- we find the average velocity between a time interval so small that it is thevelocity at an instant.

Since we have shown that the velocity at time a is simply the velocity between a time interval as that

interval becomes such an extremely small quantity, that it approaches 0. Therefore the velocity at aninstant is, once again,

= lim 0 =


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Since instantaneous velocity is velocity at an instant, in order to calculate an "instant", the time elapsedapproaches 0. So if we take the limit as the time interval approaches 0, to get the time interval at aninstance, we derive instantaneous velocity.

Now that we have discovered average velocity and instantaneous velocity, scientists stumbled upon anew and revolutionary concept - acceleration (well all discoveries are "revolutionary"). Velocity causes achange in position but what causes a change in velocity? Since empirical and observational data showsthat there are very few ideal situations where velocity remains a constant, the mysterious borders andembodies the change in velocity. What if we take into account the change in velocity (if is there)? Let usexamine Alpha, if you remember, who drove a car through a freeway and a stop sign, so his velocities atthose instances (or any other for that matter) are not necessarily the same. Alpha was driving at a highervelocity during the ride throughout the freeway rather than driving to a complete stop at the stop sign.For at one instance, Alpha could have been driving at some velocity, but at another Alpha could driveanother velocity. So the question must be asked- how do we account for this change in velocity in our sofar developed kinematics? This is what acceleration defines, that is, the rate at which velocity changes.

So acceleration is logically and mathematically expressed as

= this simply means that the acceleration is the rate at which velocity changes. Since the internationalunits of velocity is meter per second, and acceleration is velocity over time:

=

To further dimensionalize acceleration, meter is a measurement of length and second is a measurementof time, so it follows that

=[ ]

[ ]

[ ]

=[ ][ ]2

If the acceleration of an object is simply 1 meter per second squared, then by the definition of acceleration-

= 1 1 This simply expresses that after every 1 second, the velocity of the object has increased by 1 meter persecond.


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If the acceleration is now 2 meters per second squared, then it follows similarly that

= 2 1 after every second, the object increases its velocity by 2 meters per second.

To further elaborate the contemplation of acceleration, scientists expanded on the prime concept of acceleration

= = Since the is simply the change in velocity or and let simply be now denoted as whichboth represent the time elapsed, we form a new equation

=

= + In essence, the actual velocity of object at a latter instant or at a final instant, depends on the velocity of the object before the acceleration and after some amount of time.

So if a car is travelling 10 meters per second, then hits on the gas pedal which accelerates the car at anaverage 5 meters per second squared, what is the velocity of the car after 10 seconds? Since the velocitybefore the acceleration (the initial velocity) is 10 meters per second, and the car accelerates an averageof 5 meters per second squared for 10 seconds, the final velocity can be deduced from our derivedequation.

= + = 10 + 5(10)

= 60

So the velocity of the car after it accelerates is 60 meters a second.

As from before in the position vs time graph, the slope told us the velocity, and likewise, in a velocity vstime graph- the slope tells us acceleration. Since slope of any arbitrary function ( ) is simply( ),If ( ) represents velocity and represents time, then the slope of this particular function is whichthe exact definition of acceleration as mentioned before is.


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Notice that the change in velocity over the time elapsed is the slope of the velocity vs time graph and is

therefore the average acceleration over that time interval.

Yet, also as investigated before, the acceleration is not necessarily the same at every instant. In this caseof this graph, observe that the average rate of change of velocity is a linear approximation over sometime interval. Yet if I want to capture the instantaneous acceleration or the change of velocity at aninstant, then the time interval of an instant approaches 0. So the instantaneous acceleration is aninstantaneous change in acceleration which is (with similar reasoning as instantaneous velocity):

= lim 0 If, in the previous graph, we want to find the instantaneous velocity at time t_i, then we must find the

slope at the instant t_i. And in order to find the slope at the instant t_i, we must lessen the time intervaluntil, eventually, t_f approaches t_i. So if we lessen the time interval somewhat, then

The slope of this lesser interval is still just the average acceleration over an interval. So if we want to findthe acceleration at the instant of t_i, then we must lessen the interval to 0, so the slope is theacceleration at an instant. Further lessening the time interval:


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But notice that the interval does not yet approach 0, there is still some delay so no longer are findingacceleration at an instant, but rather the average acceleration over this small time interval. So if we

literally let the time interval go to 0, then we find:

Then notice that the acceleration at an instant is simply the slope of a velocity vs time graph at aninstant. This proves the correlation between velocity and instantaneous acceleration which is howscientists have derived and how we understand instantaneous acceleration-

= lim

0

=

So we find the acceleration at an instant when the time interval approaches 0, because an instant has 0time interval, and then we can find the instantaneous change in velocity over an instantaneous change


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in time which will give us the instantaneous acceleration.

Though there is apparent relationship between acceleration and velocity (by definition acceleration isthe rate at which velocity changes over time), and there is also apparent relationship between velocityand displacement (by definition velocity is the rate at which displacement changes over time), the

question arises: what is the relationship between acceleration and displacement? If there is accelerationacting on a body, then we know that the body changes its velocity. If the body changes velocity, then weknow the object has changed its position. Yet how can scientists calculate and logically derive theamount of displacement based off acceleration? What is the relationship between acceleration anddisplacement? In order to make physics much easier, scientists presumably assumed that acceleration isa constant. If velocity is a constant, then it would be easy to calculate displacement over certain timeinterval, so scientists used constant acceleration to find the displacement over a certain time interval. Tounderstand its mathematical and theoretical relationship, let us first start with constant velocity. If velocity remains a constant that acts on some object, then the velocity does not change over time, it is aconstant. So a respective velocity vs time graph would be shown as follows:

Since velocity is some constant v we know that the average velocity is just this constant v .

= since the average velocity is just the velocity (since velocity is constant in whit case),

=

= Notice that if velocity is a constant, the change in position is simply:

( )= = ( )+


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If we substitute constant for v and we let be t = ( ) +

So the displacement changes based from time interval so the displacement can be rewritten as afunction of time:

( ) = ( ) +

where the ( ) simply means the final position as a function of time (this does not mean x multiplied byt ). However notice the beauty of this formed equation- since velocity is a constant and is also someconstant, we can consider this equation as:

= +

which is an equation of a line.

Since represents the final position with respect to a certain amount of time (or a position to timegraph), and is a constant which is the slope of the graph (as velocity is the slope of a position vs timegraph) and b is just some y-intercept, which is just the initial position of the object. Basically, if thevelocity is a constant, then the rate at which the position changes is linear. So in a position vs timegraph, if velocity is a constant, than the slope is a constant which means this graphs out to be some line.

and notice that when the time is 0, or when this graph crosses the y-axis means the initial position of the

object. Also note that the definition of displacement is simply the cage in position- so in this case if onewanted to calculate the change in position in this case:

= =


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However, let be replaced by t as they both simply mean the time elapsed. Also, since velocity is aconstant, the average velocity does not change; in other words, the average velocity is that constantvelocity v . So substituting these concepts in, we find that the displacement of an object if velocity is aconstant is

=

Though this is just the restatement and re-analysis of the past- as we have probed into the matter of velocities' relationship to displacement, this helps us deduce acceleration's relationship to displacement.If acceleration, rather than velocity, is a constant than what might the displacement be? Let us beginwith the definition of acceleration:

= = But since the acceleration is a constant, the average acceleration is that same constant so let us call thatacceleration a . Also let us rewrite the as just t because the signs are rather messy in our logicprocess- t and both represent the variables of the time elapsed.

= The means change in velocity which is logically the final velocity minus the initial velocity as thismathematically tells us how much the velocity has changed (hence, change in velocity).

=

= +

Furthermore, since a is just a constant, we can rewrite this equation with this equivalence-

= ( ) +

To even fully demonstrate, let us take create a function of velocity to time as the final velocity changesas the time interval changes.

( ) = ( ) +

where the v(t) refers to the velocity as a function of time. If we were to substitute the constant

acceleration as m and as b (since is a constant that does not change as time changes), we get theform:

( ) = ( ) +

This is the equation of a line, as the slope is a constant- so if we graph a velocity to time graph, if acceleration is a constant, the slope of the velocity vs time graph is a constant which means that thegraph is linearly shaped.


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( ) = ( ) +

Since there is some constant acceleration, the velocity changes at a constant rate. But then how can wemeasure the displacement since the velocity changes at a constant rate because of the acceleration?Only when velocity remains a constant, does displacement change at a constant rate, but when velocity

changes at a constant rate how much the object does displace? So let us consider this thoughtexperiment: assume there is an object - a box- stranded alone in space. However this box is moving at aconstant positive velocity which means that the box is moving at a constant speed to the right. Then if some observer were to capture instances of the box while it moves at fixed constant time intervals, thenlet us visualize how these instances, overlapped, might look. In other words, let this box be moving at aconstant velocity (to the right) and let us take pictures of the box after a certain constant time interval.Since the box has velocity, it has obviously changed its position- that is, the box has displaced. And if wewait a certain constant amount of time to record the objects position, in which the box has a constantvelocity, the object displaces a constant interval.

So if object at some instant 1, were to have some constant velocity, then after some amount of time the object moved its position. We then record this position after that amount of time which we willcall instant 2. Then after another of that same amount of time, the object changed its position thesame as it did before because it is travelling at a constant velocity and we will call this instant 3. Timeafter instant 3 is what we will define instant 4 and the object has traveled the same distance as betweenall the other intervals because the object is travelling at an instant velocity for a constant time. So if wewere it take this box at instant 1 and then take a picture, we get just the box at instant 1. If we then takea picture at instant 2, interval after instant 1, we see the box has moved by some distance. If we takethe picture at instant 3 se see that box has traversed that same amount of distance and so on and soforth all because the velocity is a constant. If now in the another experiment we want to take these"pictures" and frames of instances for the object's position if the object is traveling at constantacceleration -

At instant 1, let us say the objet has an initial velocity of zero but has some positive constantacceleration *which means that the object gains velocity as time passes by). So then after some time,called instant 2, the object has traveled some distance because of the acceleration. But because the


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velocity keeps increasing every moment of time or every instantaneous frame, the object covers moreand more distance. From instant 2 to 3 (which is still the same time interval), the object now does nothave an initial velocity of zero. Since the object has gained velocity from instant 1 to instant 2, the initialvelocity at instant 2 to instant 3 is the same velocity as instant 2. So now the object has some initialvelocity and has content acceleration so at every point in time the object increase its velocity at some

constant rate which means the object travels faster and faster. So even after the same amount of time, the object converse a greater distance from instant 2 to 3 then instant 1 to 2 simple because thevelocity is much higher. However from instant 3 to 4 the object covers even more distance since thevelocity is even greater than those before (since this positive constant acceleration causes the velocityto increase at a constant rate). Finally, from instant 4 to 5, the object traverses the most distance thenthe other former instances since the velocities are even higher. Though acceleration is a constant, thevelocity is not as by definition of acceleration, the velocity must change. Since in this case the velocity isincreasing and increasing, the object travels a greater and greater distances. Though it is apparent andonly seemingly logical that under constant acceleration, an object must cover more and more distance,the inquiry is made on what exact value is this displacement under constant acceleration? So thisrelationship must be sought by us learners in order to understand the everlasting quest of truth.

If acceleration is a constant value, then in acceleration versus time graph:

Since it is a constant, acceleration remains the same after any amount of time. As derived before, if acceleration is a non-zero constant, then the velocity versus time graph must be linear whichmathematically can be expressed as

= ( ) +

Then since a is some constant acceleration, the velocity to time graph becomes a line with a constantslope (or constant acceleration).


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To further demonstrate, since acceleration is the rate at which velocity changes and hence the slope of

velocity versus time graph is acceleration, this velocity changes at a constant rate and hence the slope isa constant which defines the velocity versus time graph to be a line. Basically, at every latter instant thevelocity increases and the object travels faster and faster all since the acceleration causes the velocity toincrease. Note that the definition of average velocity is

= But with constant acceleration the velocity is changing at a constant rate so then the average velocity isnot the actual velocity at an instant. Remember that velocity is constantly increasing in this case whenacceleration is constant positive. So the average velocity is not necessarily the velocity at any instant

because the velocity keeps on increasing, in this case. But because velocity is linear and has a constantrate of change or constant acceleration, the velocity keeps on changing yet what is the average velocity?


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Since velocity is changing at a CONSTANT rate, then the average velocity follows to be right in themiddle of the initial velocity and the final velocity. In other words, the average of the velocities initiallyand finally is the average velocity throughout this interval of time.

= And since, as we have just shown, the average velocity is the average of the initial and final velocity,

=+2

= +2 () But remember from before that if acceleration is a constant, then the average acceleration is theconstant acceleration because acceleration is not changing over time.

= Substituting in average acceleration for the constant acceleration

=

= = = +

So substituting this form of final velocity back into the former equation


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= + ( + )2 () Distributing yields:

=(

) + (

)(

+ )

2

Further distributing yields:= + 2+ 2

Combining the gives= 2 + 22

Separating the division to form two separate fractions

= 2 2 + 22 And simplifying leads to the conclusive result that

= + 12 2 But in order to make this equation simpler, symbolically at least, let us call to be t as they bothassimilate the meaning of time interval-

= + 12 2 = +

12 2

As derived, this equation means that if acceleration is a constant applied over an interval t time, thenthe displacement of the object over the t time interval is the former equation (and expandingdisplacement to change in position yields the latter equation). Furthermore, since acceleration is aconstant for this equation to be true, and the initial velocity is also a constant, the only variable that

changes the final position is the time the acceleration is applied. So the position is a function of time;that is, position changes as time changes. And also note the degree of this function is 2 so this is asquared function which means if I were to graph a position to time graph if acceleration were aconstant, it would be parabolic.

If acceleration is a constant, as we have contemplated before:


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Essentially, if acceleration is a constant, then the position vs time graph is parabolic.

The equation simply means that if an object has an initial velocity of and then undergoes constantacceleration a over a certain time interval t , the object displaces over that t time interval-

= +

1

22

Therefore, under constant aeration we can relate displacement as a function of time and velocity as afunction of time, yet if in the following situation let us try to find the problems with the knownequations. Take in a theoretical experiment that there is an object undergoing a constant knownacceleration, with a measurable initial velocity, for some quantifiable amount of time. As an observer,we can measure displacement since we know , , and so we can apply them to the former derivedequation. We can also measure the final velocity of the object under these certain circumstances as thefinal velocity as a function of time under constant acceleration is as previously derived,

= +

But now let there be another theoretical experiment in which time is difficult to measure. In otherwords picture an object which is strapped on to a bomb and scientists want to calculate the object'sfinal velocity. Assuming that the bomb provides constant acceleration then we know that the object willbe undergoing constant acceleration upon the detonation of the bomb, and we also say that thisexperiment is held in a separately confined, isolated room so the scientists dont die. Then under thismanner, it would be difficult to determine the final velocity of the object as we need the constantacceleration, the initial velocity, and the time in which the acceleration acted on the object. Scientistsknow the initial velocity of the object as it is in this case 0 (the scientists initially leave the object at reststrapped on to the bomb), and scientists also know the acceleration (assuming its a constant) from

previous experiments with the bomb. But in order to calculate the time interval the scientists mustmeasure the time interval in which the acceleration is applied; that is, scientists must accuratelyquantify the time interval in which the explosion occurred but the explosion occurs extremely fast! Sothen the time interval is really small! This makes it difficult to measure the time in which theacceleration acted on the object and hence difficult to find the final velocity. But what else can scientistseasily measure in this experiment? Well scientists know the displacement in which the box travels,regardless of the object's velocity, as it traverses the same distance since the explosion pushes it withsome constant acceleration outward to the surface of the confined room. So the scientists, beforedetonating the bomb of course, could go in the confined room and measure the distance of the objectto the wall and can assume that right after the explosion, the box has a final position on the surface of

the wall. Furthermore, if the room were spherical and the box were positioned at the center, thenregardless of which specific surface on the wall the object lands (as the explosion pushes the object tothe surface of the sphere), its final position minus initial position, or change in position , (which is in thiscase the distance from the initial position of the box-which is the center- to the final position- which isthe surface of the sphere) is just the radius. Now that the displacement of the object is known, by thehypothetical reasoning above, and that the acceleration is known and the initial velocity is known (inthis case it is 0 as the object is initially placed at rest) , how can scientists logically calculate the final


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velocity of the object as it hits the surface of the sphere? In general terms, given constant acceleration,the initial velocity, and the displacement of the object, how can we relate the object's final velocity withthese given quantities? We know velocity as a function of time and displacement as a function of time,so what we want is simply velocity as a function of displacement. Lets start with the known- underconstant acceleration, the final velocity of an object is

= +

and the displacement of the object over the t time interval is

= + 12 2 Yet since the t time interval is immeasurable or unknown, then we solve for the time in the formerequation and substitute that value (which is still the time interval) into the latter equation. So solving fortime in velocity as a function of time equation:

= +

=

=

Now that we know that time can be expressed in terms of the initial velocity, final velocity, and constantacceleration, we can substitute this value for time in the displacement as a function of time equation:

= +

1

22

= + 12 2 = ( ) + ( )22 2 = ( ) + ( )22 2

=

( )+

Truth. Volume II (Proper)

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Transcript of Truth. Volume II (Proper)