Constancy

The problem with measuring the motion using displacement, velocity and acceleration is that we can’t predict accurately about the direction and length of the path at every point.

As we have discussed before in displacement, velocity and acceleration, it was considered only the two points of the motion and not all the points in the path. Displacement, velocity and acceleration defined by taking only the initial and the final position vectors of the motion. Position vectors are drawn between only these two points and it depends only on those two.
It has been never cared about “the in between path of the motion” in our definition. It is not sure about whether the motion happens in between path continuously or stops in some of the places. We also don’t know whether the motion slows down or speeds up in some places.
Take an example of travelling in a vehicle. By the way in our travel, we will slow down or speed up or even sometimes will take a break by stopping the vehicle. But from the previous definitions, velocity is defined from the position vectors of just the initial and final position. The motion is taken to be a constant one, as the position vectors and the displacement vectors are taken as straight lines having fixed magnitude of length and direction though it is not true in real.
Since, so much information about the motion has been missed out in our preceding definitions, it is then realized that these definitions are not sufficient for the complete knowledge of mechanics. It needs a mathematical tool. Aimed at the mathematical explanation of motion at every point, it started to try relating all the points in motion. And the relation should be done with our basic definitions.

Try to think it on your own on “how to relate each point in any motion using mathematics and physics??”

The task was completed by two great scientists, Isaac Newton and Gottfried Leibniz. Newton solved the task with physics and physical interpretations. Leibniz solved the same but using the concepts of mathematics.

It is fair to study about the method used by Newton in physics. The principle behind this method is simple but a little has to be understood properly.

The weakness that we are facing in our basic definitions is the matter of instantaneous measurements.
Since we know that our basic definitions are quite right, we just need to improve our methods to get complete information of any motion at every instant.

The problem is there in measurements and not in the definitions.
It can be understood by visualizing the motion as it follows in the topic creation of the term speed,

In our definitions of displacement, velocity, acceleration it is assumed that the displacement vector between two points of the motion is the straight line vector drawn between those two points of motion.

We took the direction as a constant one though the real path of travelling is curved.
 

 
And it cannot be changed the direction of a line, since the direction of a line is a constant one.

But it is started to think in a way whether it is possible to represent a curved path using straight lines.

Task: If it is possible to relate the curved path at each instant to relate with straight lines of positional and displacement vectors, then it will make the task very simple.

Solving the task with mathematics will be studied separately.

In physics it was thought in such a way as follows using constancy,

Constant means nothing but remaining same without any change.

In physics the changings are happening with respect to time. So if any quantity doesn’t change its value with respect to time then it is called ‘constant with respect to time’. In real, all the measurements are done with the help of constancy only. The time of period of this constancy can change but the constancy should happen.
For example take the sentence of saying “the speed of the vehicle is 10 meters per second” though it is not constant for the whole motion. From this sentence we can be damn sure that the vehicle is achieving the constant speed of reaching 10 meters per one second at some instant. But we don’t know whether it stays at the same velocity or not. It is also possible for that object to change its velocity at each instant. Though there are changings with time, in some amount of small time period the vehicle doesn’t change its value.

It can be easily understood from the same vehicle example. Though vehicle is changing its velocities at every instant, in the speedometer the needle crosses all the speeds. It touches a speed at least for a single moment. [Note: It is the same principle that is used in Lagrange’s Mean Value Theorem]

Though the vehicle has continuously changing speeds from 0 to 10 m/s etc. it attains constant values at every small time intervals.
Although the changing of speed is so rapid that it changes with each nano seconds, the constancy happens at every time no matter whether the change is high or low.
It can be noted above that from the 1^st nano second to 2^nd nano second it changes its speed incredibly small amount. But, in the between time period of 1 nano second the vehicle attains some constant velocity. If it is said the change happens in nano seconds, it can be argued the constancy happens in each pico (10^-12) time of intervals. If it is said the change happens in pico seconds, it can be argued the constancy happens in each femto (10^-15) time of intervals.

Hence there is always an argument of constancy in every tiny amount of time intervals.
And it was proclaimed the general mathematical notation for this kind of arguments which is called the Limits.
Instead of arguing by making the time interval so small, we made the general decrement that goes with all type of arguments which is “decreasing the time interval to zero” Or simply “decreasing the intervals to zero”.
And thus the chapter of Limits was formulated and it serves as the fundamental for differentiation and integration.

Here in physics, we used the concept of decreasing time intervals but in Mathematics the same task is finished by considering the problem of the motion in a different mathematical version with the concepts of drawing tangents to the graphs of different functions.
But don’t forget to remember our task as discussed before,

Task: If it is possible to relate the curved path at each instant to relate with straight lines of positional and displacement vectors, then it will make the task very simple.

The mathematical way is discussed completely in the chapter of functions and differentiation in Mathematics.

Change in Motion and its Units

 There are some other terms like Acceleration, Jerk and Jounce which should have been studied before the essential concepts in physics. They were improved from the other terms such as displacement, velocity in the mechanics of motion.

Those terms like Acceleration, Jerk, and Jounce can be represented very simply using mathematics but it has a matter of to be understood thoroughly in a physical way.

For instance,

Acceleration is defined similar to the definition of velocity vector as,

Acceleration = Change in velocity vector / time,

 where the

              Velocity vector = Displacement vector / time (or) Change in position vectors/ time

Since Velocity is a vector quantity, ‘change in velocity’ that is ‘acceleration’ is also a vector quantity.

  

Similarly,        

                                    Jerk                 =          Change in acceleration vector/ time

and,                            

                                   Jounce             =          Change in Jerk vector / time

Note: You can ask whether it can be defined further as Change in Jounce vectors and etc.

Yeah it can be defined. But it is enough to describe most of the motions that happens in the universe using these terms. If you want to define, you can do. But it is up to Nature’s choice whether it is going to be used or not in real life problems.

There are given certain units for all these mechanical terms. It is called system of units. SI units are the most widely used unit system.  

As per the SI units,

                                                the fundamental unit of            length is          meter

and                                          the fundamental unit of            time      is          second.

Using the fundamental units all other units can be derived using simple arithmetic.

            Displacement is a vector that has only length and direction.

It should be noted that directions are always symbolized by writing separately near the physical quantities.  Therefore,

Unit of “displacement” is meter

and also                          Unit of “distance” is meter,     as they both are the measure of lengths.

Since displacement is measured in meters, change in displacement is also denoted using meters.

Unit of velocity vector = Unit of “change in position vectors”/ Unit of “time” (or)

Unit of “displacement vector”/ Unit of “time”

gives,                           Unit of velocity = meter/ second

and also                       Unit of speed = meter/ second,          as they both are the measure of rate of change of lengths.

Similarly, Unit of acceleration is derived as,

                       

Unit of Acceleration vector = Unit of “Change in velocity vectors” /Unit of “time” (or)

Unit of “Velocity vectors”/Unit of “time”

→                                Unit of Acceleration vector =  (meter/second)/second

→                                Unit of Acceleration vector =  (meter/second)* 1/second

→                                Unit of Acceleration vector =  meter/(second^2)

Correspondingly,        

Unit of Jerk vector = Unit of (acceleration vector / time ) = meter/ (second^3)

Unit of Jounce vector = Unit of (Jerk vector / time) = meter/ (second^4)

Thus Units were given for most of the physics terms using SI units.

Now, it was come to the situation of being a little bit saturated in the definitions of mechanics. And so now on, it will be discussed real life problems that can be solved using the above concepts. Also it will be discussed about the mathematical improvements that were made on the calculations of displacement, velocity, acceleration, etc.

            In future it will be seen that, those mathematical improvements made a very great evolution in the history of mathematics, called the differentiation and integration developed by the great scientists both Isaac Newton and Leibniz.

Concept of Velocity

 The term speed was used to measure several physical things. As it is insufficient only speed does not give full details about any motion.

By the term speed we can get information about the fastness of the approaching of any object but it doesn’t tell any information about the direction. Without direction, it cannot be understood completely the motion at each and every moment in space.

From the means of knowledge about the speed of an enemy army, we cannot decide the way by which the attack is going to happen. Suppose if the king travels with his army in the direction completely opposite to the direction of enemies, then it is completely a nuisance.

We know that, motion is “the change in orientation of the point in space with respect to time”.

The orientation of any moving object can only be pointed out, only if it was known about the distance and the direction of that object from a fixed reference point.

It was able to get the information of distance from the term “speed” though it is not adequate.

Besides to get a clear understanding of motion, it was started to designate new terms and measurements to define direction of any object.

           

The Direction of any motion or anything is completely indicated with

“The Cartesian Co-ordinate system”. 

As Direction is the one of the fundamental concept for the understanding of physics and mathematics, it was separately studied and analyzed in a detailed manner in the chapter of “Vectors”.

And that is how it was created two new kinds of quantities in physics so-called Vector and Scalar quantities.

Vector quantities are those which have both magnitude and direction. It gives information about both the number value and the direction of any physical quantity.  

Subsequently, Scalar quantities are those which don’t have any direction but only the magnitude.

E.g. Speed is a scalar quantity, which has only magnitude. Similarly distance and time is also a scalar quantity.

From the definition of Vectors, we can conclude Motion is a vector thing as it changes with direction. Hence to define Motion, we need vector quantities.

            Already the most successful term that defines is motion is Speed. If Speed could be defined in a vector form then it might be possible to get information about Motion in the simplest form. 

            Henceforth Speed defined in a vector manner by giving “direction to the distance” travelled by the object.

Since “time” has no direction in mechanics, “direction to the distance” gives “direction to the speed” by way of the formula,

                        Speed (s) = Distance (d) / time (t)

In addition, it was named new terms for the quantities that have direction. And the new names are,

Displacement (r)          -           distance that has direction

Velocity (v)                  -           speed that has direction

To point out the difference between vector and scalar quantities,

            Vector quantities are written by putting a small arrow mark above the top as

But for the sake of simplicity in our text format, it will be often written all vectors by its name and not by the sign.

e.g. 

Displacement vector    –          vector r
Velocity vector             –          vector v

The introduction of vectors made slight changes in the measurements of distance from the previous measurements as it was done in speed.

When coming to directions, it needs to be careful to note that “direction is represented only by the straight lines of rays and not by the curved ones”.

Though the motion happens in a curved path, displacement is measured using only straight lines of rays.

In any motion straight line of rays are used to represent the direction of any point and that ray is called the position vector of the moving point at that instant of time.

Since Motion is the change in orientation [change in position], Displacement is calculated by the change in position vectors of the initial and the final points.  

It was not considered the whole path of the motion but the initial and the final.    

The “change in position vector” is called the displacement vector and it is illustrated by drawing a ray from the initial to the final point of the motion.  

•          The numerical size of the ray is measured to be the magnitude of the displacement vector.
•          The direction of that ray with respect to a fixed reference direction is called the direction of the displacement vector.

In most of the physical problems, horizontal direction is taken as the fixed ones and all other directions are measured from this horizontal. Where it can be seen in Cartesian Co-ordinate system, the fixed horizontal direction is known as positive X - axis.

Thus it was able to measure both the magnitude and direction of any displacement vector using positional vectors.  

             If you take the motion as from the origin O to point A, then the displacement vector is denoted simply as vector r and it is directed θ angles from the horizontal.  

            See Figure: Displacement vector of a motion that has initial point at origin O

Being the initial point is at origin, position vector of initial point is zero. And so, it was given only one positional vector which is “the positional vector of the final point – vector r”.

That is why it is not obvious to see the change in position vectors in the picture of motion that is given below figure. 

But if you take the same motion in some other reference frame that is situated somewhere else in the space, then it need to be drawn two rays from a new fixed point O` to the initial and the final point of the motion.

See Figure: Position vectors of a motion

The two rays are the position vectors of initial and final position O and A.  

And the direction of the position vectors of these two points is measured by the angles θ1 [theta 1] and θ2 [theta 2].

           

But here, the displacement vector is measured using the positional vectors.   

The ray joining the initial and the final position tells the direction of the motion by which the point object moves. And the magnitude of the joining vector is defined as the magnitude of the displacement and its direction is defined as the direction of the displaced motion.

Thus displacement vector is the joining vector line that connects the position vectors r1 and r2.

From the vector sum rule it is known that

           vector O`O   +   vector OA                    =    vector O`A

          vector r1       +   displacement vector r   =    vector r2

                      displacement vector r   =    vector r2 – vector r1

Hence the displacement vector is written mathematically 

as and it makes “theta - θ” angles from the horizontal.

The angle theta made by the displacement vector with the horizontal position can be deduced by the angles of the initial and the final positions.

            Depending on the motion, position vectors changes its directions and magnitude with respect to time.

           Thus speed was further redefined with direction using the newly created term – displacement.

And it is called the velocity which is used often in physics.

Velocity is defined in a same way how the speed was defined. Instead of the distance term, it was replaced by the displacement term.

     Velocity vector (v)      =     displacement vector (r) / time (t)      

(or)

Displacement Vector ( r ) = Change in position vectors, therefore,

           

Velocity vector (v)  =  Change in position 

                                   vectors  / time (t)

It should be noted a relation between speed and the velocity. It was known that, displacement is the line drawn between the initial and the final points of the motion.

From the chapter of geometry in mathematics, it was known that the shortest distance between any two points in the space is the length of the line that connects the two points.

Henceforth displacement is the shortest distance between any two points of the motion that is considered.

            All other distances [lengths] that measured between those two points should be equal or greater than the length of the displacement vector.

            So, with correct mathematical words it is stated as,    

Magnitude of displacement vector is always less than or equal to magnitude of the distance of the path in any motion

Magnitude of displacement is the numerical value of the length of the displacement vector.

To represent the magnitude of any quantity it will be regularly used the symbol | |   [drawing two lines on both sides of the physical term].

The statement can be written as,

      |displacement vector|          ≤          |distance of path|

  

Dividing a scalar quantity will not affect the above relation and so divide by time on both sides

→   |displacement vector / time|   ≤      |distance of the path / time|

 →     |velocity|                             ≤      |speed|

Thus magnitude of velocity is always less than or equal to speed.

Note:

Writing the same statement as,

           displacement vector       ≤          distance of the path

is wrong because,  

    “Displacement vector” is a vector that has both magnitude and direction but

    “Distance of the path” has only magnitude.

      

      Comparison can be done only between same kinds of properties.

An apple cannot be compared with a mango in mathematical calculations. That is why it was compared only the magnitudes that is common for both these physical quantities.

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