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.