While playing with a ball, Nora gets curious
about its motion.
As she drops the ball on the floor, she asks
herself “What will be it's speed as it reaches
midway in it's path?”
She drops the ball from a height one metre
above the ground.
It covers 50 centimetres to reach the mid-point.
Nora knows that it took one second for the
ball to reach the midpoint B. With this information,
can she find the speed of the ball exactly
when it’s at point B?
Like you’d probably be thinking, Nora also
thinks the speed of the ball will be the distance
travelled by it divided by the time taken
to reach that point.
So she comes up with the answer ‘50 centimetres
per second’ or zero point five metres per
second.
But is this the speed of the ball when it’s at point ‘B’?
No, it's not.
This answer would have been correct if the
speed of the ball was constant throughout
it's motion.
But we know that the speed of the ball increases
as it falls.
So, the answer Nora got is actually the AVERAGE
speed of the ball as it reaches position ‘B’.
But what we are interested in is the speed
exactly at the INSTANT when the ball is at
position ‘B’.
That is called the instantaneous speed of the ball.
Can you try finding the instantaneous speed?
Let’s see what happens at the instant the
ball is at position ‘B’.
The distance travelled by the ball at this
instant is zero and the time elapsed at this
instant is zero.
So we get the speed to be ‘zero divided
by zero’ which is undefined.
Doesn’t make any sense right!
How do we then find the instantaneous speed
of the ball?
CALCULUS is the branch of mathematics that
helps us answer this question.
How?
We will see that in the later section of this course.
But wait… another thought puzzled Nora.
As she drops the ball, she wonders, why the
ball ever reaches the floor…
This might seem to be a lame thought, but
don’t forget that Nora’s smart.
She thinks that mathematically, the ball should
never touch the ground.
So,what was her thought process?
Let’s see!
Suppose she drops the ball from a height one
meter above the floor.
Now to reach the floor, first the ball has
to cover half this distance to reach point
‘B’.
Then the ball has to cover half of the remaining
distance; that is one fourth of a meter.
Then the ball has to cover the next half;
then the next half and so on.
It means the number of steps the ball has
to cover to reach the floor does not end.
That is, there are infinite number of steps
the ball has to perform.
And to perform these steps, the ball takes
an infinite amount of time.
So according to this logic, Nora thinks the
ball requires an INFINITE amount of time to
reach the floor.
Therefore the ball should never reach the floor right?
Do you also think the same?
Do you think Nora went wrong somewhere?
Share your thoughts in the comments section
Actually, Nora isn’t only the one who was
puzzled by this.
Many centuries ago, the same thought puzzled
a Greek philosopher, Zeno of Elea .
This is usually referred to as Zeno’s Dichotomy paradox.
Even though we know that when we drop the
ball it reaches the floor, this logical and
mathematical conclusion tells us that it should
never reach the floor.
Again, a satisfactory answer to the Zeno’s
paradox is provided by Calculus.
We saw two examples here that calculus can
give us the answer to!
But before looking at the central ideas of
calculus, we will further explore what other
real life problems calculus can help us with.
If we are on a cliff next to the sea, it’s
always tempting to randomly throw stones into
the sea.
It’s so much fun right!
But have you ever wondered about the best
possible way to throw a stone such that it
covers the maximum distance?
Knowing this was certainly important in the
past to attack the enemy’s ship.
Now,let’s get back to our question.
If we throw a stone too high, we know it will
not cover maximum distance.
What if we throw the stone horizontally?
Maybe not!
By experience, we know instead of throwing
the stone horizontally, if we throw it at
an angle, it will cover greater distance.
Of course the answer also depends on the speed
with which you throw the stone.
Let’s say, if you apply all your energy,
you can throw it with a speed ‘V’.
So if we throw the stone with a fixed speed
‘v’, at what angle should we throw it
to cover maximum possible distance?
As the angle at which we throw the stone changes,
the distance covered by it changes.
And this is where calculus comes into play.
To get the answer we need to know how the
distance covered by the stone changes, as
the angle we throw it at changes.
And this is exactly the kind of problem that
Calculus helps us with.
Alright, so calculus helps us with analysing
things in motion.
For instance, finding the instantaneous speed
of an object, or finding the angle at which
to throw the stone.
But wait…
Let me ask you a completely random question.
Look at this trajectory of the stone.
What do you think will be this area under
the dashed curved path?
We know how to find the area of a simple shape
like the rectangle.
It's area is equal to its length times its width.
But how do we get this formula?
Let’s say the length of the rectangle is
‘5’ centimetres and its width is ‘10’
centimetres.
Then the area of the rectangle is fifty ‘square
centimetres’.
So,what does this mean?
It means that if we take a square tile of
length of one centimetre; that is a square
tile of area one SQUARE centimetre, then fifty
such tiles will cover this rectangle.
Now let’s get back to our question.
What will be the area under this curve?
Should we cover this area also with square tiles?
This will not work right!
Look at the square tiles covering the curve.
We have a problem here as they don’t fit perfectly.
Then how can we figure out this area?
You would have guessed by now that calculus
helps us to find the answer.
We know the area of simple shapes like rectangles,
triangles, polygons and so on.
Here are the formulas!
This is easy because straight lines are involved.
But the shapes that we encounter in our daily
lives are not that simple, as curves are involved.
That’s where calculus comes into the picture!
So we have seen that other than finding the
instantaneous speed of an object and the angle
at which to throw an object to cover maximum
distance, calculus also helps us to find the
area of different shapes.
In this course about Calculus, we will explore
each of these examples in detail.
But before moving on, let’s have a glimpse
at the central idea around calculus.
This idea was used by Greek mathematicians,
to find the area of a shape, long before calculus
was developed.
Consider this Circle with radius ‘r’.
How would you find its area?
Consider these two triangles: one circumscribed
around the circle, and the other inscribed
inside it.
We can say that the area of the circle will
be between the areas of these two triangles.
Now what if we used squares instead of triangles.
We will get a better approximation of the
area of circle if instead of triangles, we
use squares.
We can further improve our results if we used
pentagons.
Did you get the idea?
Can you tell me how we can improve the approximation
further?
As we consider polygons with greater number
of sides, we will get close to the circle.
The area of the polygon inscribed in the circle
and the area of the polygon circumscribing
the circle get closer to each other.
This was the method used by Greek mathematicians
to find the area of circle.
It's called the method of exhaustion.
This is the central idea of Calculus used
to solve the problems we mentioned above.
With this knowledge, do you think we can solve
our problem of finding the instantaneous speed
of an object?
Think about the ways in which you can approach
the problem and share your thoughts in the
comments section.
In the next part, we will see how to find
the instantaneous speed of an object and the
idea applied to calculate the area of a shape.
We will also discover that these two ideas
are related to each other.
See you in the next part!
