February 28, 2018 Wednesday
Bedtime Story
Breaking Down the Brachistochrone Problem
The brachistochrone problem au contraire,
in spite having such a complex sounding name, is extremely simple to define and
grasp by any sapiens ape.
In fact, it is so childishly simple that
most apes even assume that even its answer would be intuitively obvious.
And that is where they fall flat.
Jacob had initially planned to wait till 6
months before publishing his own solution in the journal but during that period
he got no response.
Then at Leibniz’s request he extended the
response period to a year and a half.
Eventually Johann got responses from 5
mathematicians, namely Newton, Leibniz, his own elder brother Jacob,
Tschirnhaus (German name pronounced as Shianhaus) and de l’Hopital (French name
pronounced as de Lopital).
Johan himself offered not one but two
solutions, one that he called direct and the other indirect method.
Before I go to these masters, it is
important to realize that the most intuitive answer to the brachistochrone
problem by most average apes would be a straight line joining point A to B.
I shall try to make it intuitively clear
(without going into the mathematics of it) as to why this answer is incorrect.
We have to keep in mind that in the
solution to this problem we need to utilize the one and the only force acting
on the particle to its maximum efficiency: the acceleration due to gravity or
the gravitation force.
Since the time also depends on the
distance, we also have to minimize the distance that the point needs to travel.
Let us simplify the problem and consider
the point A to the on the tip of the y axis and the point B at the tip of the x
axis with both the axis meeting at point (0, 0).
Now surely the distance is shortest when a
line directly joins the point A and B.
But the frictionless point that would roll
down this line would not be maximizing the use of gravity’s acceleration as
intuitively we know that an object falling vertically will derive maximum
acceleration.
An accelerating body continues to increase
its velocity with ever second and in a state of falling body or in case a
rolling body, the effect will be more the steeper the inclination.
But if we follow the other extreme path and
let the point drop down directly from A to the point (0, 0) and then let it
roll down the horizontal path, then this path would be tediously long and would
take the away the benefit of the acceleration.
Hence what we do need is something
intermediate that would give the best of both the worlds, meaning maximum
acceleration with the shortest distance that allows it.
There is something in nature that utilizes
this method all the time as it itself reaches from one place to another in the
shortest time possible.
You are very familiar with this object and
also with a great deal of physics related to it.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
