Wednesday, 11 August 2021

PROJECTILE MOTION SIMPLY EXPLAINED WITH EXAMPLES VISUALLY.

 

So, you are a visual learner and want to learn about projectile motion intuitively huh ?


The projectile motion of an object has 2 components.

1) The velocity component parallel to the ground, aka

real part of the velocity aka 

initial velocity x cos(angle of throw)

Denoted as

Vo x cos(thetha)

(Initial velocity in some books is denoted as u, in others as Vo

I will be using Vo. ) 


2) Then there is a perpendicular component to this velocity aka the imaginary part of the velocity . 

Written as

Initial velocity x sin(angle of throw) 

Denoted as 

Vo x sin(thetha)

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An analogy to this would be a single branch of a tree splitting into 2 branches.

Or taking an axe and cutting a log of wood into 2 parts.Is it necessary that these daughter parts should be of equal size w.r.t each other ? But surely bringing them together will create the whole right ?

Similarly (real part)^2 + (imaginary part)^2 = (whole thing)^2.

So squaring and bringing together the real and imaginary parts via addition will give you the whole thing.









Now coming back to projectile motion,








 

 

 

Range actually = (2 x real part of initial velocity x imaginary part of initial velocity) / acceleration due to gravity.




Thursday, 28 January 2021

PURPOSE OF DETERMINANT OF A MATRIX IN REAL DAILY LIFE .

 

SO WHAT IS THE PURPOSE OF THE DETERMINANT OF A MATRIX IN REAL DAILY LIFE ? 

A Determinant is a measure of how many times larger is the area (in case of a 2 x 2 matrix) or Volume ( in case of a 3 x 3 matrix) w.r.t the area or Volume a unit cube.

So Determinant = RATIO OF how much is the area of the object matrix under consideration divided by the area of a unit square.


 

For 3 dimensional figures (3 x 3 ) matrices, instead of area ------> substitute volume. And instead of unit square ---------> substitute area.

BUT FIRST, I WANT YOU TO KNOW ABOUT HOW MATRICES ARE MAPPED ONTO A GRAPH.



 


 


 

 

Every element of the matrix may represent a certain co-ordinate that the matrix is stuck upon or hung upon.

For example, just look at the matrix above.What you will see is that the number 3 in the left-most and top-most section of the matrix represents 3 on the x-axis. 


3 lies on the x -axis and there is no y and z component to it. This means that there is one point or one vertex of the matrix such that it has co-ordinates of (3,0,0) on a 3-D graph.

 KEEP SCROLLING BELOW FURTHER THIS PAGE TO READ ABOUT NEGATIVE VALUES OF DETERMINANTS .......






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 






 

 

 YOUTUBE VIDEO

 
 
 
 
 
APPLICATION OF MATRICES IN REAL LIFE VIDEO


RELATED LINKS 

RELATIONSHIP BETWEEN DETERMINANT AND VOLUME OF A MATRIX


Determinants of a matrices give a relative volume of a matrix as compared to the volume of a  unit matrix. 

This unit matrix will be a unit square if you are dealing with say a 2 dimensional matrix.

This unit matrix will be a unit cube if you are dealing with say a 3 dimensional matrix.

WHAT IF THE DETERMINANT OF A MATRIX IS -VE ? 

Say you get the determinant of a matrix as -2 .

It still means that the volume of your matrix or the area that your matrix is trying to denote is 2 times that of the unit cube or unit square.

THE ABSOLUTE VALUE OF YOUR MATRIX IS WHAT COUNTS
  
Not whether it is positive or negative.


 


Saturday, 23 January 2021

INVERSE OF A MATRIX WITH APPLICATION AND PURPOSE IN REAL DAILY LIFE






 


Analogy / Example for inverse of matrix

Imagine a software through which if you put an image(digital), this image stretches in width by a factor of 4

and its height increases by a factor of 3/2 .

 

Now lets say you want to make a software that has to bring the image back to its original dimensions.

This software does this by multiplying the stretched and increased height modified matrix by a new matrix such that the resulting matrix will

be the original unstretched and unmodified image.

 

The parameters of the unmodified image can be represented as Matrix x [ Identity Matrix] .

Just as every in numbers, every value can be represented as number x 1,  For example your weight can be represented as 65 x 1.

In a similar manner in the world of matrices, every matrix can be represented as Matrix x (Identity matrix).

The number 1 when multiplied to any number means an operation which does not change anything about the number. Its actually multiplication

by the fraction 1/1. We just loosely write it as the number 1. Its actually a fraction.

 

Now imagine you are going in a rocket and suddenly your rockets speed increases by 4 times. And you need some mechanism to slow the rocket down

to how it was before the power surge happened. You would require some software that will reduce the speed of the rocket by 1/4.

As you see, 1/4 is the inverse of 4.

 

Now lets say its a car that has dented due to an accident. Certain dimensions of the car have dented inwards by a factor of 1/1.2 

The inverse of 1/1.2 is 1.2

This entire incident of denting and undenting can be represented by matrix multiplications. That matrix which will produce  the undenting and

recover the original dimensions of the car is known as the inverse matrix.






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 






 

 

Thursday, 14 January 2021

VISUALIZING FLUX AND FLUX EQUATION INTUITIVELY.



EXAMPLE OF FLUX

Flux is any effect or influence like heat or magnetic field or electric field which usually has the ability to pass through substances and surfaces.

Flux describes how much of an influence is passing out through a surface. 

Eg:- X - Rays,

 

ANALOGY FOR FLUX

Hair growing from your scalp or grasses growing from the ground. The denser they are, the more you speak of flux density.

 

PARALLEL AND PERPENDICULAR COMPONENTS OF FLUX

 

 

Flux has a component perpendicular to the surface. And it has another component parallel to the surface. Only the component perpendicular to the surface has a significant effect . This component perpendicular to the surface will be parallel to the normal 'n'. The normal is a line drawn perpendicular to the surface.

So the line that is perpendicular to the surface will be parallel to the normal. Because the normal itself if an imaginary line perpendicular to the surface. So any flux line perpendicular to the surface will be parallel to the normal and these are the only lines which have any significant

So another way of putting it is that only that flux components which is parallel to the normal ( and  thus perpendicular to the surface ) will have any significant effect .

 

EQUATION OF FLUX

It is  Flux strength experienced = How strong the field intensity is  x How large the surface area is of the surface through which flux is passing x how parallel the normal (an imaginary perpendicular line drawn to the surface is ) x how parallel the field lines are w.r.t the normal drawn to the surface (this angle is represented by cos(thetha)

The above explanation in formula format is as.....

F = E.A.cos(angle)

 

FLUX IS ZERO WHEN FIELD IS PERPENDICULAR TO THE SURFACE ( And thus parallel to the normal).

ANALOGY FOR THIS IS AS FOLLOWS

Now imagine that this arrow is being shot northwards. It will hit the target because the target seems to face it.

But now imagine an arrow being shot from west to east or vice-versa. Will this east-west arrow hit the target at all. No, right ? Because it runs parallel to the plane of target (bullseye plank ).

 

COS( ) COMES INTO MATH WHEN 2 FORCES/INFLUENCES ARE SUCH THAT THEY PRODUCE MAXIMUM IMPACT WHEN KEPT PARALLEL TO EACH OTHER.

So, if in the equation , you see cos( ), that means that net forces will be maximum when the influences are parallel to each other.


 

 

 

 

 

 


SOURCES

WHY ELECTRIC FLUX IS ZERO IF ELECTRIC FIELD IS PERPENDICULAR TO SURFACE

 






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book

 

  

 

 

 






 

Tuesday, 29 December 2020

APPLICATIONS OF COMPLEX NUMBERS OR IMAGINARY NUMBERS IN ELECTRONICS

                   Whenever a force/influence etc gets split into two daughter forces/influences or two branches,...complex numbers come into the picture. 

In electronics, when you put a resistor in the path of electrons, all that they do is burn electrons. So there is only one effect the overall circuit has on the army of electrons, burning.

But now when you put a capacitor or an inductor in the path of the incoming electron army, what do they do ?

 They delay the flow of electrons. They as if act as a traffic police or say a dam. They hold electrons for some time and then they leave these electrons .

So now there are two daughter branches of effects on the incoming army of electrons. 

1) The resistance which fries electrons .

This is also called as the real part.

2) The capacitance which just delays the electrons.

This is also called as the imaginary part.

Yes, I too am searching for the guy who came up with the nomenclature of real and imaginary parts . One day I will get him in my hands.

 






















e-book links

VISUALIZING MATH 1 e-book link


 

 

 

 

 

 

 

 

 

 

 

PDF LINK FOR VISUALIZING MATH 2
https://gum.co/visualizingmath2book