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

__Speed and Velocity__

This is the first Physics lecture, where a solid mathematical background will be needed.

In particular, we assume the familiarity with

**vectors**and

**derivatives**.

These and many other mathematical concepts can be found in the course

"Math 4 Teens" on the same site as this lecture - UNIZOR.COM.

In the first approximation

*speed*is a distance covered by a moving object in a unit of time.

What's wrong with this definition?

Firstly, a moving object moves with a different speed at different

moments in time, so speed is a function of time, which is not reflected

in this definition.

Secondly, what is a "unit of time"? Any unit of time, like a second, is

not a "moment" in time, but an interval, during which an object can

still change its speed.

Let's approach this concept more rigorously.

Our task is to define a speed as a characteristic of motion at any moment in time.

The simplest motion we consider first is a movement of an object along

the X-axis, according to the following function of X-coordinate of time

**:**

*t*

*x(t) = a·t*Let's define the

*average speed*during any time interval from moment

*to moment*

**t**_{1}*as the distance covered during this time interval divided by the length of time this distance was covered.*

**t**_{2}For an object moving along the X-axis this

*average speed*can be expressed in a formula:

**[**

*s*

=_{ave}(t_{1}, t_{2}) ==

**]**

*x(t*_{2}) − x(t_{1})

*/ (t*_{2}− t_{1})Remarkably, the average speed for an object uniformly moving along the X-axis as

*x(t) = a·t**and*

**t**_{1}*- the beginning and ending moments in time:*

**t**_{2}**[**

*s*

=_{ave}(t_{1}, t_{2}) ==

**]**

*a·t*_{2}− a·t_{1}

*/ (t*_{2}− t_{1}) = aOf course, this type of movement is called

*uniform*exactly for this reason -

*average speed*of movement is constant on any time interval.

Consider now a movement along the X-axis according to general function

**and the task of defining its speed at any moment**

*x(t)***.**

*t*Let's note the position of our object at two moments in time -

**and**

*t***Δ**

*t+***. We can find an**

*t**average speed*during this time period:

**[**

*s*

=_{ave}(*,***t****Δ***t+***) =***t*=

**Δ**

*x(t+***]**

*t) − x(t)***Δ**

*/ (t+***[**

*t − t) =*

==

**Δ**

*x(t+***]**

*t) − x(t)***Δ**

*/*

*t*It is reasonable now to define a speed at moment

**, called**

*t**instantaneous speed*

**, as a**

*s(t)***limit**of the above expression for the average speed on interval [

**Δ**

*t, t+***], when Δ**

*t***is diminishing to**

*t***, that is time increment Δ**

*0***is an infinitesimal variable:**

*t*

*s(t) =*

==

*lim*[

_{Δt→0}**Δ**

*x(t+***]**

*t) − x(t)***Δ**

*/*

*t*The limit above is a

**definition of the derivative**of a function

**. Therefore, the rigorous definition of an**

*x(t)**instantaneous speed*of an object moving along the X-axis according to function of time

**is the**

*x(t)***of this function:**

*derivative*

*s(t) =**d*

**x(t)/**d**t = x'(t)**In one of the previous lectures we mentioned that coordinate functions

**,**

*x(t)***and**

*y(t)***must be**

*z(t)**continuous*to prevent instant science fiction jumps to other planets. From now on we will also assume that these coordinate functions are

*differentiable*to be able to determine the speed of movement as a derivative of a coordinate function by time.

This completes the definition of

*speed*for a one-dimensional movement along the X-axis.

Now we will address general case of movement in three-dimensional space described by coordinate functions

**,**

*x(t)***and**

*y(t)***.**

*z(t)*To define a speed in case of general three-dimensional movement represented by three coordinate functions

**,**

*x(t)***and**

*y(t)***, we will use the vector representation of the position of an object.**

*z(t)*So, assume that at time from

**to**

*t***Δ**

*t+***a vector**

*t***from the origin of coordinates to point**

*a***,**

*x(t)***,**

*y(t)***}**

*z(t)***from the origin of coordinates to point**

*b***Δ**

*x(t+***,**

*t)***Δ**

*y(t+***,**

*t)***Δ**

*z(t+***}.**

*t)*Then the difference between these vectors is a displacement that

happened with a moving object during this time. The first important

thing to notice here is that this displacement is a

*vector*

**.**

*d = b − a*Though we did not mention the vector character of the movement along the

X-axis above, it was implicitly there, because movements forward

(towards increasing X-coordinates) or backward (towards decreasing

X-coordinates) are two different directions that are characteristic of

vectors. So, the positive or negative displacement

**Δ**

*x(t+***]**

*t) − x(t)***Δ**

*x(t+***and negative if**

*t) > x(t)***Δ**

*x(t+***, and a sign in one-dimentional case of movement along the X-axis is the direction.**

*t) < x(t)*Now, when we have determined the vector character of a displacement from a position

**at time**

*a***to position**

*t***at time**

*b***Δ**

*t+***, we can define the**

*t**average displacement*during this period of time - it's a

*vector***Δ**

*(b − a) /***.**

*t*Coordinates of this vector are, obviously,

**Δ**

*x(t)/***, Δ**

*t***Δ**

*y(t)/***, Δ**

*t***Δ**

*z(t)/***},**

*t*Δ

**Δ**

*x(t) = x(t+***,**

*t) − x(t)*Δ

**Δ**

*y(t) = y(t+***,**

*t) − y(t)*Δ

**Δ**

*z(t) = z(t+***.**

*t) − z(t)*When time increment Δ

**is infinitesimal, all three components of the displacement vector are infinitesimal as well.**

*t*If all three coordinate functions are differentiable (which we will always assume), the limit of the

*average displacement vector*

**Δ**

*(b − a) /*

*t***, when Δ**

*t***, is a vector with coordinates**

*t→0***,**

*x'(t)***,**

*y'(t)***}.**

*z'(t)*Hence, the

*instantaneous speed*of movement of an object defined by coordinate functions

**,**

*x(t)***and**

*y(t)***is a**

*z(t)***with coordinates**

*vector***,**

*x'(t)***,**

*y'(t)***}.**

*z'(t)*This vector is called

**, while the term**

*velocity***is reserved only for the**

*speed**magnitude*of this vector, that is its absolute value, disregarding the direction.

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