Solution to The Two Trains Puzzle

As promised I am going to post a solution to the two trains problem.
I restate the problem below:

Two trains are speeding towards each other on the same track both at a speed of 50 km/hr. At the instant they are 100 km apart Bumble Bee starts flying from the windscreen of the first train to the second train at a speed of 75 km/hr. When it reaches the windscreen of the second train it bounces back at the same speed to the other train. It continues repeating this cycle indefinitely. Determine the total. distance traveled by Bumble Bee until the instant it experiences an untimely death due to the catastrophic head on collision between these trains.

And now for the solution:

Here is a visualization of the problem:

And here is the solution:

First we take the instant the two trains are 100 km apart.
Bumble Bee will travel towards the second train and the second train will travel towards Bumble Bee for a time 't' until they meet. The distance travelled during this time must be 100 km.

75t +  50t = 100 
125t = 100
t = 4/5

Now Bumble Bee travels towards the other train but this time the trains are only 20 km apart. So:

75t + 50t = 20
125t = 20
t = 4/25

Now Bumble Bee again travels towards the other train. But this time the trains are only 4 km apart. So the time until they meet will be 4/125.

What I am trying to do is to find the total time Bumble Bee spends bouncing between the trains. If I have that figure I can multiply it by the speed of Bumble Bee and get the distance. If you glance at the time you can immediately comprehend that the time keeps reducing but does not become zero unless I repeat the process of finding the time an infinite number of times. Serendipitously there exists a method of finding the sum of all those times that is better than spending an infinite amount of time adding an infinite amount of numbers.

We can express the total time T as an Infinite Geometric Series.

T = 4/5 + 4/25 + 4/125 + .................... ad infinitum
or 

T =

It takes only a moron to realize that this Infinite Geometric Series converges to 1. So the total amount of time Bumble Bee spends bouncing between the trains is 1 hour.

The total distance traveled is therefore 1 x 75 = 75 km      
                                                                       ______
                                                                       --------
And that is a truly beautiful result.

Winter Solstice

It is the day of the Winter solstice, the day on which the Earth is tilted as far away from the sun as possible. And so today at midday the sun would have been tilted away form normal by as much as 23.15 degrees at the equator..

Predictably the northern hemisphere will experience it's shortest duration of daylight today. Today was also the day I planned to recheck my calculation of the tilt of the Earth's axis. But alas! The weather is unaccommodating. Despite the fact that it was reasonably bright int he morning, just before noon, clouds started gathering ominously in the sky and any hopes of a measurement crashed.

Another Calculation

Today I decided to do another calculation of interest related to the Earth. 

I chose to calculate the speed of rotation of the Earth. I needed some reference point and therefore chose to calculate the speed of one point on the Equator. I considered  the day of the Equinox i.e the day on which the duration of daylight and night time would be exactly the same(12 hours). I imagined myself on a desert island in the sea which is located exactly on the equator. On the day of the Equinox the sun would travel 180 degrees from the west to the east. From this assumption I obtained a crucial figure: The angular velocity of the Earth. Since 180 degrees in pi radians the angular velocity would be pi/12 radians per hour.

The radius of the earth is 6367.5 km. The velocity of one point on the Equator would therefore be (pi/12) * (6367.5) = 1674.365 km/hr or 463.05766716974556587756670701464 m/s.

But that was easy I did not even have to make any physical measurements. I decided to make a very interesting calculation: The difference between the speed of my feet and the speed of the top of my head. I state the radius of the Earth in meters: 6367500 m. my height is 1.67 m. 

The speed of the top of my head is.  463.05778861......(I am not displaying all 32 digits in my calculator display but I use all the digits for my calculation.)

Subtracting the speeds I find that my head is 0.0001214458271179387663942735177012 m/s faster than my foot. So during the day my foot only travels   40008182.443466016891821763486064 m while my head travels 40008192.936385479881731179951293 m. Subtracting, I get that my head travels 10.5 meters more than my foot in one day!!

Now I apply Einstein's Relativistic equation for time dilation:

The relative speed between my head and  my foot or v is 0.0001214458271179387663942735177012 m/s.

c is the speed of light. Let t-zero be 1. 

I get t-v to be  1.0000000000000000000000000820529 seconds

This means that for every second that passes for my head 1.0000000000000000000000000820529 seconds pass for my foot.

The average life expectancy of the world is 65 years. If I live to be 65, my head will have lived a total of 2049840000 seconds while my foot will have lived a total of 2049840000.000000000000000168195316536 seconds.

This means that at the end of 65 years my foot will be older than my head by 1.7 * 10^-16 or  

0.000000000000000168195316536


seconds!!

I think you would agree that this is a truly astounding revelation.

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A Delightful Little Problem

Here is a tricky little puzzle:

Two trains are speeding towards each other on the same track both at a speed of 50 km/hr. At the instant they are 100 km apart Bumble Bee starts flying from the windscreen of the first train to the second train at a speed of 75 km/hr. When it reaches the windscreen of the second train it bounces back at the same speed to the other train. It continues repeating this cycle indefinitely. Determine the total. distance traveled by Bumble Bee until the instant it experiences an untimely death due to the catastrophic head on collision between these trains.

It shouldn't take more than five minutes and requires only a basic understanding of mathematics and a bit of logical reasoning. I invite the readers to try to solve this puzzle and post your solutions in the comments below.(A word of advice: Make sure YOU do the problem and please try to do yourself justice by abstaining from searching for a solution online.) I will be posting a solution to this problem in one of the following posts.

A Few Clarifications.

In my entry before the last there was a lot going on behind the scenes which I didn't explain. For the benefit of my scientifically inclined readers I am going to explain the the chain of reasoning I went through to arrive at the final answer and why my answer was as accurate as it was. One of my friends had sent me an email with an inquiry of  how I reasoned it out. Here I present my reply. I hope it clears all doubts.

To do this calculation I had to make an assumption.

First of all, I had to assume that the earth is tilted as far as possible away from the sun. Since It is December Which is winter time in the northern hemisphere, this is a pretty good approximation. But I couldn't be sure. However something else partially cancels this effect(namely, the latitude of my position).

I reasoned that since it was only the 8th of December, the earth could not have reached this maximum position(In actual fact, maximum is reached on December 22). So the angle that I calculate must be smaller than 23.45 degrees. I the checked the location of the city I stay in and I found out that it was on the 5 degree North latitude. So I reasoned that even if the angle would be smaller this smaller angle I get would be larger than the original tilt by five degrees (Since I'm 5 degrees above the equator). So if the earth was not at this maximum position, my angle would be made larger by roughly 5 degrees and my answer would therefore be closer to the correct answer.

And most importantly I made this calculation at exactly 12 noon, a time when the sun should have been directly above my head making an angle of 90 degrees. If I made the calculation at this time, then any deviation suffered by the sun from normal must be the tilt of the earth.

So I have taken into account the Latitude (5 degrees greater), and the time of the day. I do not need to worry about the longitude because all places on the same longitude will experience midday at exactly the same time. I think that covers everything.

If Zero by Zero were defined....

If zero by zero were defined a whole lot of crazy things would start to happen.
 Just as an example I quote this section of Wikipedia.

With the following assumptions:

The following must be true:

Dividing by zero gives:

Simplified, yields:

The fallacy is the implicit assumption that dividing by 0 is a legitimate operation with 0/0 = 1.

Although most people would probably recognize the above "proof" as fallacious, the same argument can be presented in a way that makes it harder to spot the error. For example, consider the following equations:

Dividing by x − x gives:

and dividing by x gives:

Therefore it would  be possible to prove that 1 = 2 and 2=3. In fact it would be within our capability to prove that any number in the number system is equal to any other number. If this were true then everything would be exactly the the same.It is therefore obvious that division by zero is undefined. If zero by zero were defined all the buildings and bridges would tumble and crumble. All calculations would be rendered useless. You wouldn't be able to tell the difference between an elephant and an ant. If zero by zero were defined, there would be no difference between you and me, no difference between big and small, no difference between hot and cold. The universe would be astoundingly uniform. There would be no stars or galaxies or beautiful nebulae. No black holes, no Quasars, no interstellar dust. The universe would be just one big boring entity.

The Tilt of the Earth's Axis

We are all aware of the fact that the Earth's axis is tilted from the normal. This tilt is the cause of the seasons.  Today , to relieve myself of boredom I decided to calculate this tilt of the Earth's axis using a very tall lamp post and its shadow both of which could be observed from my bedroom window.

                                                                             The Tilt
I started with the shadow. I measured the length of the shadow using my normal 6.5 inch (15 cm) ruler. It recorded a length of 3.5 cm. Careful to hold my head so that it's spatial co-ordinates remain reasonably constant I measured the height of the lamp post. I knew that all angles on the triangle I constructed(See illustration below) would be exactly the same as those in the triangle formed by the original lamp post and shadow because I was working with similar triangles.

 So I did the math.
Theta would be the arctan of (8/3.5). 
θ = 66.4°

The angle of tilt (φ):
φ = 90 - 66.4°
φ = 23.6°
________
-----------

The actual angle of tilt of the Earth's axis is 23.45°. The error in my calculation is therefore 0.6% which is absolutely fantastic. 


And that is the story of how I measured the angle at which the Earth's axis tilts away from the normal.

“The important thing is to not stop questioning. Curiosity has its own reason for existing.”

—Albert Einstein --- Quoted by William Miller in Life Magazine, May 2,1955

The Magic Square

I took a break from revision today after noon by constructing a 19 x 19 magic square. It turned out to be the most beautiful thing I created by hand. Behold!

It has numbers from 1 to 361 and it took exactly 20 minutes to build. That was better than I had expected. After all, I had to fill up the matrix so that each row column and diagonal gave a sum of  3439.
If you refuse to believe it I invite you to download the picture, enlarge it and utilize a calculator (or your brain) calculate the sum.

Truth

Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.

—Isaac Newton

The Beauty Within

It is undeniable that nature is beautiful. Rainbows, Auroras, Lightning, Waterfalls...... and on goes the list. I have on many occassions experienced this beauty for myself but only recently have I realised that all this beauty that I experienced was just on the surface. I was missing out on a lot of things. When I looked at a rainbow I only saw the colours and the pleasing pattern in which they arranged themselves.

Over the holidays I read extensively on electricity and magnetism. In the process I learnt a lot about some natural phenomena that I never gave much thought about.

Lightning is one example. I now know exactly how lightning is formed in the clouds - not just about the charge on the clouds and all that but the nitty-gritty details. I then realised that the experiences of the people without this knowledge are limited. They don't see the true beauty, the beauty within.

Another example is the rainbow. From now on whenever I see a rainbow I will always verify that the colour red is indeed on the outer side of the rainbow and blue on the inner. I will always compare the section of sky inside the rainbow and the sky outside and confirm that the inside is always brighter than the outside. I will not be able to resist looking for the secondary bow outside the first one and observe that the colour sequence is inverted as compared to the brighter innner bow. It is a disease for which there is no cure. And of course I will always know that the light is very strongly polarized. I will then walk away from this experience enriched and enlightened knowing that I have experienced the beauty within the rainbow, the invisible beauty that very few have experienced.

Ah the joy of learning Physics!

Celebrate!

The world's largest particle accelerator ot the CERN is due to reusme operation in november this year.

The LHC (Large Hadron Collider) was shut down soon after it's first test run in the september of 2008 due to a technical problem. One of the superconducting magnets which control the path of the particle beam had a small problem. Fantastic amounts of current flow throuogh these superconductors and when this current encountered a small resistance it produced an extremely large amount of heat damaging a section of the LHC and other superconducting magnets nearby.

These magnets have been repaired and the tests are almost complete. The LHC will only start off with particle beams of lower energies (3,5 TeV).

Perseid Meteor Shower

The Earth is entering a stream of particles emitted by the Comet Swift-Tuttle. You can expect metoer showers to light up the skies on August 11 and 12.

What causes meteor showers?

A meteor shower (some of which are known as "meteor storms" or "meteor outbursts") is a celestial event in which a number of meteors are observed to radiate from one point in thenight sky. These meteors are caused by streams of cosmic debris called meteoroids entering Earth's atmosphere at extremely high speed on parallel trajectories. Most are smaller than a grain of sand, so almost all meteoroids disintegrate and never hit the Earth's surface. Fragments which do contact Earth's surface are called meteorites.

The Skeptics Sociey, Michael Shermer

Murray Gell-Man

Brian Greene - The Universe on a String:- A Talk On the String theory

Ode on a Grecian Urn - By John Keats.

The poem really made me think. We all perceive beauty in different ways. I saw that in class yesterday. But what in reality is beauty? Does it exist because we exist to observe it? Or does it have an existence independent of an observer? Is there something which has a 'Universal Beauty' to it such that all things percieve it as beautiful?

Many people seemed to find nature beautiful. I suppose that fact is irrefutable. But in my opinion what is truly beautiful is mathematics. It intricately and subtly connects everything in this universe. I think all mathemetics is abundant in beauty but if told to choose just one thing in it that I find beautiful I always go for Euler's Equation

e^iπ =cos π + i sin π

This I think is a unique fromula because it manages to relate five fundamental numbers into one elegant equation.

Free Will Does NOT Exist.

Mr Raj, this is in reply to your comment on my previous blogpost. The reply was so long that I thought it would be better to put it in a separate blogpost instead

But no matter what decisions we make it is a result of electrical impulses in our brains. And these electrical impulses are a result of the movement of Sodium+ ions in our brain. Irrespective of how we make our decision we still can't make the Sodium+ ion move in a way that defies the laws of physics. When you choose to marry someone the "love" you feel corresponds to specific areas of the brain firing electrical impulses. By using Newtonian Mechanics slightly modified with General Relativity and the Principles of Quantum Mechanics we can calculate the fate of all the Sodium ions that created the electrical impulses. That is the theory put in its simplest form. If we ant to count the external factors, we take into consideration the events that happen around us. For example, choosing a career. It is a few specific events in a person's life that leads a person to choose a career. Other factors include mental ability, opportunities and finances. Mental abilities are completely determined by the size and health of specific sections in the brain which consists of atoms whose fate can be determined. Opportunities are determined by the state of the economy and other factors which are affected by the import/export of goods and specific economic events. All these events include matter which is made up of atoms whose fate can be predicted. Finances mean money and other assets which can be liquidated into cash thus helping you to get some education thus helping you to choose the desired career. Since money and assets are made up of matter they contain atoms whose fate can be predicted. As for events in a person's life those events are a direct result of interaction between matter particles (if the event involves you father telling you something his mouth is affecting the air molecules to move which carries the sound to your ears which make your eardrums move causing electrical impulses in your brain causing a decision/opinion to be formed.) I could also explain with appropriate detail how any other factors affect the decision making process. Give me any event ant I'll explain it. As difficult as it may seem to believe, all events ARE predetermined by the laws of physics. The notion that we are free to act as we wish is an Illusion. In fact we are forced to act as though free will existed by the Laws of Physics. Because, In theory we can map every single movement of each and every particle in the universe right from it's creation about 380000 years after the Big Bang up to it's destruction(Whenever that is). If we can do that in the atomic level it follows that we can do that in the macroscopic level. The only reason that we haven't done this yet is because as the size of the system we are dealing gets larger, the number of equations which we deal with gets larger and larger. So to predict the fate of a system as large as the human body we would nee computers with at least a hundred (or more) times the computing speed of today's computers. According to Moore's law (Which states that power of computers doubles approximately every 18 months. And it has stood wonderfully till now) that will take another 50-100 years. But stepping stones to the creation of such 'quantum computers' of immense power have already been laid. Scientists have used the power of 7 atoms vibrating exactly in phase with each other to calculate that 5 x 7 = 35. Although this is simple, we may see exceedingly complex quantum computers as the years advance.

Free Will - Does it Really Exist?

 "To Newton and Einstein, the notion of free will, the notion that we are masters of our own destiny was an illusion. The commonsense notion of reality, that concrete objects that we touch are real and exist in definite states, Einstein called "objective reality." He most clearly presented his position as follows:

I am a determinist, compelled to act as though free will existed, because if I wish to live in a civilized society, I must act responsibly. I know philosophically a murderer is not responsible for his crimes, but I prefer not to take tea with him. My career has been determined by various forces over which I have no control, primarily those mysterious glands in which nature prepares the very essence of life. Henry Ford may call it his Inner Voice, Socrates referred to it as his daemon: each man explains in his own way the fact that the human will is not free . . . Everything is determined . . . by forces over which we have no control . . . for the insect as well as for the star. Human beings, vegetables or cosmic dust, we all dance to a mysterious time, intoned in the distance by an invisible player." - From: "Parallel Worlds", by Michio Kaku

The above statement may seem more related to the concept of God than Science, but scientifically speaking, this statement holds good. Because, theoretically, if we calculate the fate of every particle in a human body taking into account all the forces acting on it we are effectively predicting the fate of that person! The problem is, that the number of forces acting on one particle alone are so numerous that we would need a computer with computing ability millennium ahead of the current level to calculate the fate of a system as large as a human body.

This theory however leads to a most startling conclusion. It implies that the fate of every single particle in the world is predetermined and can be predicted right up to it's destruction. Effectively this means that all events that happen in this world, from the motion of stars and galaxies right down to the beat of a butterfly's wing and your neighbour picking his nose was predetermined by the Laws of Physics at the time of the Big Bang! 

     

The Strange World of Quantum Physics - Heisenberg's uncertainty Priciple.

A few days ago, I came across a very interesting effect of Heisenberg's Uncertianty Principle.

The theory states that it is impossible to know simultaneously with an infinte degree of accuracy both the position and velocity of a particle. In a very beautiful and exquisite way the diffraction pattern one observes in light is one of the proofs of this theory. 

Imagine that we are shining a light through a large slit so that there is a beam of light falling on a screen.

Now imagine that we are slowly making the slit smaller and smaller. The iwdth of the beam will get smaller ans smaller. But afte a certain point Heisenberg's Uncertainty Principle comes into action. since we know the position of the light  beam with greater and greater accuracy, we will know the velocity and therefore the momentum with lesser and lesser accuracy(Since P=mv). This results in the beam of light widening out and displaying a diffraction pattern as follows:

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Michio Kaku, the author is a renowned Theoretical Physicist who frequently appears in the Discovery science channel. 

The Music of Numbers

Something that I discovered a few days ago just proved that mathematics and nature are closely and inseparably entwined with each other. 

It all started when I developed an interest for a certain sequence called the Fibonacci Sequence. I'd been fiddling around with it for quite some time when on March the 7th one brilliant idea struck me as I was waking up in the morning. Why not relate the sequence to the musical notes  A B C D E F and G? I immediately grabbed a piece of paper and wrote down the first few numbers of the Fibonacci sequence and started working out their corresponding musical notes. After working out the first 5 notes, I asked my brother to play them on the keyboard for me.  Astonishingly, the sequence did not create noise as I had expected but recognizable music! You can imagine my surprise; and I don't even know anything about music other than the 7 notes.  

I then proceeded to work out the rest of the sequence and six others as well, but in all of them the notes after the first five did not sound very much like music to me. 

A few days later, I told my good friend ECYL about the music sequence just to see if the music I created had any substance in it. I had yet another surprise when she told me that the whole sequence of notes represents an entire recognisable music sequence. In other words, I had made music with math. 

The very next day I ran a few searches on Google to see whether anybody had discovered the sequence before me. I felt quite elated after 1 hour of searching revealed nothing alike. I was beginning to think I had made a really big discovery, but I was wrong. After a few well worded searches in Google Scholar, I unearthed a long lost research paper on Fibonacci Pitch Sets written by someone called Casey Mongoven. 

I was a bit disappointed at first, but then I realized the implications of what had just happened. I had actually discovered something a university scholar had all by myself! It just proves that I have the right mindset for research. 

So I am now searching for another topic to research on.  

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The Curious Sequence

A few days ago during my mathematical investigations, I discovered something of an anomaly. 

A normal Arithmetic or Geometric progression can be accurately represented by a formula which can be used to find the (n)th term of that progression. In this particular situation, I was working on the Fibonacci Sequence in which each term is the  sum of the previous two numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 .......................................

Now I thought it was a normal geometric progression and applied the formula to find out the (n)th term. First I divided successive terms to find a definite ratio between the numbers. To my surprise, there was none. I then tried to find the formula  thinking that it was an arithmetic progression. I tried to find the difference between the terms, and to my utter surprise found that each time I tried to find the constant difference between, a new Fibonacci Sequence is formed. Amazing, isn't it? 

The Recipe

The previous week I concocted a cake recipe for my English assignment. I was about to do what all other's would do: copy and paste the essentials from the internet and add their own touch. Then an idea struck me. I decided to do it impromptu, and just write down what came to my head. This was the the result.

1.Combine the butter, water, sugar, cocoa powder and chocolate together in a medium sized saucepan. 

2. Place over gentle heat. Bring to the boil whilst stirring. You need to stir until everything has dissolved to stop the mix from burning. After stirring 7 times clockwise, add 12 ml of pure honey of the finest quality. Continue stirring anti-clockwise for 13 minutes. Be careful that the mix does not boil over. Simmer gently for 5 minutes. After the third minute add three drops of vanilla essence and sandalwood extract.

3. Remove from the heat. Cover with a lid and leave to cool to room temperature. Heat the mixture gently again. Using a thermometer to measure the temperature, follow the next step while trying to keep the temperature constant at 37°C/98.6°F.  

4. Whisk in gently the flour and sodium bicarbonate into the liquid. Whisk 144 times. All lumps should have dissolved into the batter now. The batter should have achieved the viscosity of a mixture of corn-flour, soy-bean, sun-flower and olive oils.  

5. Pour batter into the prepared baking pan. Remember the baking pan must be lined with linseed oil mixed with unsalted butter. The batter should level naturally in 30 seconds. Tap the tin on the kitchen bench 8 times to remove any large air bubbles. 

6. Bake in a preheated oven set at 169.975oC (337.955°F) for 88 minutes. The top of the cake will crack and display a poison ivy pattern. Don’t worry the cracks will settle and be hidden when you turn the cake out of the tin. But Internal fissures will form, dividing the internal into three layers. This will not collapse as the mix is quite rigid. 

7. Remove from the oven. Leave in bright sunlight or a UV-Lamp for 11.5 minutes. The cake must display a colour of light brown tinged with golden patches.

8. During the 11.5 minutes melt 35 bars of 350g Toblerone chocolates in a vessel big enough to hold the cake but of small enough width to make the chocolate liquid level above the cake. Use a thermometer to keep the solution at 50°C. Gently drop the cake into the mix. It should sink completely within 45 seconds. 

9. To add to the flavor heat until the chocolate starts bubbling and within 5 minutes, add essence of strawberry and a teaspoon of Asafoetida powder. After 5 minutes the cake will have completely absorbed the chocolate and will sit in the container with a 0.8 mm covering of chocolate on the surface and two 1.3 cm layers inside.

The Time Paradox

Whenever the talk of time travel occurs, a phenomena that is sure to be the topic of discussion in the trusty time paradox. A time paradox is explained by the most common example: the "Grandfather Paradox". In this paradox you go back in time to kill your own grandfather and hence you wouldn't exist in the future and so you wouldn't have been able to go and kill your grandfather........(The sentence runs off as much as the effect does.). This explanation was invented to explain some of the concepts of time travel that we did not quite understand.

In my opinion the time paradox does not exist. We cannot change the past. If in the future I go back to change the past in some way, then we are already feeling the repercussions of my actions. If I go back I've already been back. This leads to the one definite statement: We cannot change our past or present, we can only change our future. But however if we do an action in the past which affects my going back to the past then we go into another parallel universe in which the results of our actions can happen. 

That is another implication of time travel. It implies that there are many millions of parallel universes in which the many million possibilities of that could happen, happen. In the case of the grandfather paradox the person who kills his grandfather is transported to another universe in which this twisted reality can happen. At least that is my understanding.

There are many methods of time travel that exist only in theory.On very radical theory is the transport of information through wormholes or space shortcuts. Any objects with mass cannot travel faster than light or at the speed of light. So they cannot travel through wormholes unharmed. The idea is to send streams of information through this wormhole to beam ourselves to the past. We convert ourselves into a stream of information that the time tunnel will reconstruct into our bodies at the end. According to theory information cannot be destroyed as it travels through the time tunnel or wormhole.

Another method involves using circular beams of high energy lasers to bend the space-time fabric. According to Einstein, sapce and time are closely related, interwoven and interlocked like a fabric. If space is bent enough it will warp time with it allowing us to slow down or speed up time.

The world of science is interesting indeed....