Tuesday, April 10, 2018

Unizor - Physics4Teens - Mechanics - Kinematics - Motion

Notes to a video lecture on http://www.unizor.com


When talking about motion, we have to start by defining two things:

what is moving (moving object) and

what does it mean that this object is moving.

These are qualitative characteristics.

When these issues are clarified, the next issue is to describe the
characteristics of the moving object and to describe in certain terms
the parameters of the movement.

These are quantitative characteristics.

Then, knowing quantitative characteristics of the moving object and the
parameters of its movement, our task is to come up with certain Laws of
Motion. These laws are supposed to be universal and independent of the
characteristics of the moving object and parameters of its movement.

Let's examine the moving object. In this course in most cases we will assume that the moving object is a geometric point in three-dimensional space. Therefore, the size of a moving object is zero in all directions.

Before defining the motion, we have to introduce another physical concept - time.

First of all, time is an undefinable characteristic of all processes taking place in our world. The fact that time
is undefinable is not a good news, but the good news is that we can
measure it by comparing any process with some standard process and
calculating the rate of any process in units of comparison of it with
the rate of a standard process.

Since our moving object is a point in three-dimensional space, we can
consider some system of coordinates in this space and talk about a position
of our moving object (point) in terms of its coordinates. So, three
coordinates of our moving object in this system of coordinates determine
its position.

Now we can define a concept of motion as a process (the rate of which can be measured by time) during which the position of our moving object (a point with three coordinates) is changing with time.

If M is a moving object, we can talk about its position at any moment of time t defined by three coordinates - functions of time x(t), y(t) and z(t).

We have come up with a conclusion that motion can be represented as three functions of time - the X-, Y- and Z-coordinates of a point that represents our moving object.

Since any function, in mathematical terms, represents a transformation of one number (in our case, time) into another (in our case, coordinate), we have to understand what kind of numbers (time arguments and coordinate values) we are dealing with.

Let's start with time.

To talk about numbers that represent time, we have to know how to
measure time. We need to measure the time interval, so we need to know
when the time starts, which in many cases is a moment the motion starts,
and how long it lasts from one moment (say, from the beginning of
motion) to another moment (for example, a moment of observation). That
requires the unit of time to measure this interval.

Let's assume that we have chosen an interval of time of one rotation of our planet Earth around its axis to be equal to 24 hours, each hour to be equal to 60 minutes and each minute to be equal to 60 seconds. So, the interval of time of one rotation of Earth equals to 24 hours or 24·60=1440 minutes or 1440·60=86400 seconds.

From now on in most cases we will use seconds as the main unit of
time because this is an international standard. Having the unit of time
defined, we can talk about the numerical time argument in coordinate
functions x(t), y(t) and z(t).

Now let's consider what values these coordinate functions take.

This is a familiar Cartesian coordinates, so all we need is a system of
coordinates. But this is exactly the problem. Which system of
coordinates should we choose and which unit of length to use?

The answer to this question is not easy. Under typical conditions we can
say that the system of coordinates has its origin at the point of the
beginning of the motion, which means that coordinates of our moving
object (point) at time t=0 are equal to 0, that is x(0)=0, y(0)=0 and z(0)=0.

As for direction of the axis of coordinates, we might choose any three
orthogonal directions, but usually will choose them to simplify the
coordinate functions.

For example, if we are dealing with a straight line movement, we better
choose X-axis along the direction of this movement, which leaves only
one coordinate function, x(t), to represent the movement, since other coordinate functions, y(t) and z(t) will always be equal to zero for any time moment t.

Finally, the unit of length should be chosen, and there are many. In most cases we will use metric meter as a unit of length along each coordinate axis, because it's an international standard.

With a choice of the beginning of time, the unit of time measurement,
the coordinate system and the unit of length we have fully defined all
the components needed to meaningfully describe the motion as represented by three coordinate functions x(t), y(t) and z(t), where time t is an argument and function values define the position of a moving object (point) in our three-dimensional space at time t.

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