www.mathcentre.ac.uk/.../Substitution_Formulae.mp4

A formula is a recipe or a rule
for doing something and it works

every time it gives the
relationship between different

quantities. Some formerly a
standard formally written down

by Mathematicians and scientists
to cover a wide range of the

relationships between different

quantities. Others can be made
up such as that for a washing

powder or a magician's memory

trick. More of that later.

Formally, usually use variables
and letters instead of numbers,

and that gives the relationship.
Let's have a look at.

The formula for the area
of a circle A equals π R

squared, where a is the
area of the circle.

And R is the radius of
the circle.

This formula works every time
and shows the relationship.

So we square the radius. We
multiplied by pie and it

gives us the area and it will
always work.

Imagine we have a circular lawn
where the radius is 3 meters.

And we want to know how much
turf in square meters that we

need to order so R is equal to 3

meters. So what we're going
to do instead of writing are

in our formula. We're going
to write the number 3, so A

is equal to Π * 3 squared.
Now, it's really important

that we put the
multiplication sign back in

as soon as we put numbers
back in our formula.

Π * 3 squared is π
* 9, giving us 28.3.

So the area of our.

Lawn is going to be 28.3
meters squared, so we're

going to need at least 29
meters squared of turf.

There are many formerly that
relates to the area of 2D

objects and also to the volume
of 3D solids.

Let's have a look at one. Now
let's say we want to find the

volume of a football.

Now football is a sphere, so we
want the formula for the volume

of a sphere and that's V equals
4 thirds of Π R cubed.

And let's say that the
radius of a sphere are

football is 10 centimeters.

So in our formula to workout the
volume instead of writing, are

we going to write 10
centimeters, so V equals 4

thirds π multiplied, again
remembering to put the multiply

symbol in and instead of are we
writing 10, so 10 cubed?

And if we calculate all of that.

We end up with the
volume equaling 4189, so the

volume of the football since
the radius was in centimeters

will be 4189 centimeters cubed.

There are many formerly relating
to scientific principles.

And we're going to have a look
at Newton's second law.

And that law relates force
with mass and acceleration.

And the formula is F equals
MA mass times acceleration.

Let's imagine a circus artist is
going to be fired from the

barrel. And he's going
to be fired horizontally

and the mass of a circus
artist is 60 kilograms.

And he's going to be fired
at an acceleration of 2.5

meters per second squared.

Our formula is F equals MA.

So instead of em, we're
going to write 60.

And instead of a, we're going to

put 2.5. But again, because
we're putting numbers in.

Instead of the letters, we must
remember to put the multiply

sign back in so it'll be 60
times by 2.5.

That gives us 150, so the
force on our circus artists is

150 newtons. And Newton is
a unit of force.

Let's look at an equation
of motion.

Z equals you plus 80.

Fee represents final speed.

You initial speed.

Hey, is acceleration.

Auntie is time.

And imagine that we've got some
values for you A&T.

So you.

Equal to 5.

A is equal to two and T
is equal to 3.

So to calculate V, the final
speed, we're going to

substitute these numbers
instead of these letters.

So instead of you, we write 5
instead of a. It's two, we must

write the multiply sign because
we're now putting numbers

instead of letters.

And instead of T we write 3.

Now we've got a good
opportunity here to look at

our order of operations. If
we were to start from the

left and work through to the
right, we would be in error

because we should do
multiplying before we do

addition. So a quick
reminder of our order of

operations with Bob Mass.

Where the B stands for brackets,
the apfa powers.

Steve for divide
and for multiply.

A for addition.

And S for subtraction.

So multiply comes before
addition, so we need to do

2 * 3 before we do the
addition, so it's 5 + 6,

giving us an answer of 11.

Let's look at another equation
of motion. This time V squared

equals U squared plus 2A S.

Again, the final speed.

You initial speed.

Hey Accelleration.

And S distance traveled.

And imagine we've got a Cliff.

And we throw a stone off the top
of the Cliff and we'd like to

know the speed with which it
hits the water below.

And the Cliff is
100 meters high.

So we know that you
are initial speed.

Is zero 'cause we're dropping
the stone from rest at the top?

Our acceleration is the
acceleration due to gravity, so

that's 9.8 meters per second

squared. And as the distance
that it falls is 100 meters.

So instead of the letters
in our formula, we

substitute the numbers.

UO

plus two times
a 9.8 times

S 100.

That works out at
1960. Sophie squared is

1960. So to calculate V,
the final speed when it

hits the water, we need to
square root 1960 and that

gives us an answer of 44.

And because.

Our units are meters per second
squared for acceleration in

meters for the distance that
it's fallen, the velocity is 44

meters per second.

Another equation of motion is S
equals UT plus a half 80

squared. What S is the distance?

You is the initial speed.

T is time.

A accelerations.

And the final to the same as

this one time. Not so much in
this time that we have a well.

And we want to find out
how deep the well is.

And what we do is we drop a
stone down the well.

Use the initial speed of the
stone is 0 because we dropped

it. It started at rest.

Let's say it takes 3 seconds for
the stone to hit the bottom.

And AR acceleration is that due
to gravity of 9.8 meters per

second squared. So instead of
writing you T&A in our formula,

we're going to substitute and
put these values in.

So S equals you
0 multiplied by T3.

Plus half multiplied
by 9.8.

Multiplied by T squared.

So we put all the figures in.
Now we can carry out the

calculation. 0 * 3 zero
plus half of 9.8 four point

9 * 3 squared is 9.

That gives an answer of 44 to
the nearest whole number. And

because our units are meters
per second squared and

seconds, the depth of the
world will be 44 meters.

Let's have a look at the formula
for kinetic energy.

Kinetic energy equals 1/2 MV

squared. Where M represents mass
and the is the speed that the

mass is traveling at.

The amount of work done, kinetic
energy. Let's compare a sprinter

running and the work that is
done by the sprinter running

with that of a truck. So the
sprinter is mass 70 kilos.

And the running at a speed of 10
meters per second.

Another truck

has a mass of 2000 kilos.

And that's going forward at a
speed of 20 meters per second.

So let's compare how much
work they're doing. So for

this printer.

The kinetic energy equals 1/2.
The mass is 70.

The velocity is 10 squared.

So we have 3500.

For the truck.

The kinetic energy again
is 1/2 instead of the M

we write 2000.

And instead of the V, we've got
20 to be squared.

And that works out at 400,000.

So the kinetic energy of
the truck is more than 100

times greater than that of
the sprinter.

I haven't written the units
down, but for kinetic energy

there jewels.

Let's look now at the formula
for the period of a pendulum

where T equals 2π root L over G.

What city is the period of
the pendulum?

And that means how long the

pendulum takes. To go from
one side of its motion to the

other and then back again. So
that's the period.

L is the length of the pendulum.

And she is the acceleration.

Due to gravity.

Which is 9.8 meters
per second squared.

Let's imagine we've got a
grand father Clock, and the

length of the pendulum L is
equal to 1 meter.

So in our formula.

Going to put 2π multiplied by
the square root L is 1

meter divided by G is 9.8.

And that gives us 2π. Now, if
we calculate 1 / 9.8 and then

square root the answer.

We get 0.319 *
2 and by pie

and we end up
with two .007.

So the period of the pendulum to
the nearest second is T equals 2

seconds because we've used the
units of meters per second

squared and meters.

That was a selection of standard
formerly now for the magicians

memory trick. I've got a
selection of 30 or so cards

here, each with eight digit
numbers on, and if I could have

a helper to select one at random

for me. Now, if you could
give me the two digit card

number which is on the top
left hand corner, I'll tell

you the 8 digit number on
the card number 14 #14.

OK, the
8 digit

number is

314-5943. 7, is that
right? That's correct, very

good. OK would like to try
another one, just to show that

it's not a fluke. Can you give
me the two digit number again

#13 #13? So the 8 digit number

is 29101. 123, is that correct?
That's correct, good thank you

very much. Well, I haven't
actually memorized all 30

numbers that are here. I'm using
a formula, so let's have a look

now at the numbers and show you
what I did.

Now the only information I was
given was the card number. This

number at the top left hand

corner. So I had to work out the
8 digit number from that card

number. Now the formula I
was using was 2 N at

three. What end represents
my card number?

So for example, the number 10 if
N is equal to 10.

Then I would do 2 times by 10 at
three, which gives Me 2 * 10 is

20 at 323, so that gives me my
first 2 digits of the number two

and three, and then what I do is
add the two digits to get the

third number. SO2AD3 gives me 5.

Then the next number comes from
adding the previous 2 digits.

Three at 5 gives me 8.

Five at 8 gives me 13, so I'm
going to take the 10 away and

just write down the three.

8 at three gives me 11. Again,
I'm going to take the 10 away.

And write down the one three add
one gives me four and one add 4

gives me 5, so there's my 8
digit number and all this I was

given was the card number.

OK, let's show you another one.
Let's take this one an is 6.

So 6 is going into my formula to
workout the first 2 digits, so 2

* 6 + 3 two 6 is a 12
add. Three gives me 15, so the

first 2 digits are one and five.
Then I add one and five that

gives me 6 for the 3rd digit I
add five and six. That gives me

11. I take away the 10 and one
is the next digit.

Six at one.

Gives Me 7 for the next one.

One at 7 gives me 8.

7 add 8 gives me 15. I take the
10 away, so I write down just

the Five and eight, add 5 gives
me 13 again. Take the 10 away

and I end up with three.

So there we have a magicians
memory trick. Now you can

obviously make it as easy as
complicated as you like for your

audience, so you can choose
whatever formula you want.

And delight your audience.

So to summarize.

Working with formerly.

What you do is substitute
numbers in instead of the

letters and do the calculation.
But remember the order of

operations so that you are
correct with your final answer.

And that's all you do.