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

__Motion__

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,

**of all processes taking place in our world. The fact that**

*time*is an undefinable characteristic*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

**is a moving object, we can talk about its position at any moment of time**

*M***defined by three coordinates - functions of time**

*t***,**

*x(t)***and**

*y(t)***.**

*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)***and**

*y(t)***.**

*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

**are equal to**

*t=0***, that is**

*0***,**

*x(0)=0***and**

*y(0)=0***.**

*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,

**, to represent the movement, since other coordinate functions,**

*x(t)***and**

*y(t)***will always be equal to zero for any time moment**

*z(t)***.**

*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)***and**

*y(t)***, where time**

*z(t)***is an argument and function values define the**

*t**position*of a moving object (point) in our three-dimensional space at time

**.**

*t*
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