SOFTWARE
PREVIEWS
From here you can access many
interactive routines that make up our mathematics and physics
courses.
Please feel free to experiment and don't hesitate to contact us
in case you have any questions.
Graphs of linear functions of the form y = mx + b [click]
Properties of powers [click]
Did you know? [click]
The principle of energy conversion [click]
The meaning of an increment [click]
The hydraulic press [click]
Did you know?
Did you know that very often while trying
to find the sum of first n terms of arithmetic sequence
(progression) you might apply famous Gauss formula in a
very unique way, which will allow obtaining general formula
for any amount of terms of sequence?
Indeed, let us remember what is Gauss'
formula for the sum of n terms of arithmetic sequence:
Now, let us review some simple example.
Given is a numerical sequence:
Let say, we have to find the sum of
the first 20 terms of this sequence. Classic solution of
such problem will look like this: first we have to find
the 20th term of the sequence.
Indeed, our twentieth term being 39,
while knowing, which is quite obvious, that our difference
of sequence d = 2. Once we have our 20th term in hand we
may proceed with finding our sum:
Easy enough! But what if we have to
deal with sequence in broader way? Well, there is the way.
We could derive general formula for any sum of any amount
of terms for this sequence and not to bother with calculation
of 20th, 30th or any other term. In order for us to do
that we have to remember what is explicit formula of any
arithmetic sequence (and you have to remember that explicit
formula could be written for any arithmetic sequence).
Well, this explicit formula is nothing more than formula
of the term. So, in our particular sequence it will look
like this:
Or:
You may check for yourself if this
statement is true - just plug in your n in the formula
and you will see that the first term will be 1,
second - 3, third - 5 and so on. We will get
our exact sequence, with which we started this topic. Once
the explicit formula is found, we have to return to Gauss
formula and substitute our an in it onto
our explicit formula:
Doing this, we are finding the formula
of the sum of infinite amount of terms of this particular
sequence and it is n2 , that is, we solved
problem in general form. Indeed, let us remember the sum
of how many terms of our sequence were we supposed to find - 20.
That is our n = 20. Plug this number into the general
formula of the sum and see for yourself:
Isn't this the same result we've got
by solving this problem classical way? Yes, it is! But
unlike classical way, this method allows you (once the
general form of the sum of sequence is found) to find any
sum of first n terms of sequence without even bothering
with solution for the end term of the sequence's sum. Look:
And so on. The problem, thus, is reduced
to a very simple operation.
Try to find on your own, both explicit
formulas and sums of first 10, 20 and 100 terms of next
arithmetic sequences:
Did you know that the problem of calculation
of the distance during uniformly accelerated movement could
be represented in strictly geometrical terms, or as connoisseurs
of calculus would say - could be reduced to the problem of
area, or integration? Let us review some simple derivation.
But before we do it, let us remember what is the formula
of the area of trapezoid.
What you see at the picture is a trapezoid.
The area of any trapezoid is calculated using very simple
formula:
That is, the area of trapezoid equals
half-sum of its bases multiplied by its height. Once we
recalled this simple formula, we may proceed with application
of this formula to uniformly accelerated motion. Let us
remember what acceleration is: it is ratio of change of
speed to time interval during which this change happens.
Mathematically it looks like this:
Which leads us to a simple definition
of uniformly accelerated movement:
The
movement during which for every equal interval of the
time physical body attains the same value of acceleration
is called uniformly accelerated movement. Since
we will be talking about uniformly accelerated movement
we may rewrite our equation in a simpler form, ridding
of
and
substituting it simply for
t,
since we'll be talking about equal intervals of the
time. Thus:
From here we may derive this simple
formula:
Do you understand the significance
of this derived simple relation? Well, what you see here
is nothing more than simple linear function of form y
= mx + b , where our y is v, mx is at and b is,
of course, our v0 .
Note, our acceleration a plays
the role of the slope of our linear function, and, of course,
this is true; since we are talking about uniformly accelerated
movement, which means that our acceleration is constant.
Thus we are able to represent our relation graphically
in Cartesian system of coordinates and, of course, it will
be a linear function:
Now we may analyze this
picture. Ask yourself a question: what do you see on this
picture. How
about figure OABC -
it is trapezoid, isn't it? Another matter that this trapezoid
lays on its height, which is t,
or the time during which our movement happens, while its
bases are v0 and v = v0 + at .
Thus we are facing task of finding the area of this trapezoid,
which in physics terms will be finding the distance D,
which physical body covers during time interval t,
while moving with uniform acceleration. Remember the formula
of the area of trapezoid - our distance will be the area
of this trapezoid! Using geometric formula we get:
And, indeed, the formula of the distance,
which uniformly accelerating body covers in some interval
of the time is an area of a trapezoid, which is constituted
by the initial velocity of the body v0 ,
its velocity at the end of time interval v = v0 + at ,
time interval t, which serves
as trapezoid's height and linear function, which describes
uniformly accelerated movement. In all, it is:
In simple mathematical terms, this
problem could be reduced to the problem of integration
of function v = v0 + at on
the time interval from 0 to t and
looks like this:
Try to investigate yourself,
what is going to happen when physical body starts its movement
from the state of rest, that is v0 = 0 .
What is going to happen if our body will begin to uniformly
decelerate?
