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Monday, 11 July 2016

Paradox, Confusion, Puzzle

This post is written by two of my students, I have suggested the title only…..

 

While traveling by train, both of us has gone through this experience many times…..we were in the train, the train was moving with high speed, suddenly another train comes in the side by line and it was going in the same direction of our train. We were watching the passengers of the side by train and suddenly felt that we are not moving at all, it appeared that our train and also the other train both are still. Then we immediately moved our heads on the other side and looked through the window and then we realized that our train did not stop, it was running with the same speed.

All of you have experienced it at least once while traveling by train.

So, what it says? This is a real life experience from where the concept “Relative” comes.

A moving train has a certain velocity for sure, but what is that velocity depends upon----with respect to which you measure it, generally we measure it with respect to earth, which is considered to be a “Rest Frame”---standing in the ground you measure the speed of a train(Say ‘Train A’)---obviously you will get a certain value, but if you try to measure the velocity of ‘Train A’ riding in another train (Train B)which is moving parallel to that ‘Train A’ and has same velocity then you will say that the speed of ‘Train  A’ is zero.

So, we can say everything is relative…depends upon from which frame you are measuring it.

Here some interesting incidents of Relativity have been discussed, hope you will enjoy it and will think about it….

 

Relativity:--

     We know that, the greatest scientist, Sir Albert Einstein is known as the "Father of Relativity". He informed us about relativity. Einstein formulated the 'Theory of Relativity' by which he described gravity more deeply. The theory of relativity consists of following two laws;.

 

(1) All laws in physics are same in every inertial frame; no change will take place (It is same for all non-accelerating observers).

(2.) Light has the maximum velocity (C) in the universe and it is same in every inertial frame.

Einstein made “Special Theory of Relativity’ which is generally applied where the curvature of space and time is negligible

This is what Sir Einstein said to simplify what is Relativity…

"To simplify the concept of relativity, I always use the following example: if you sit with a girl on a garden bench and the moon is shining, then for you the hour will be a minute. However, if you sit on a hot stove, the minute will be an hour."

Relativity deals with the phenomenon of ultra high speed that is the speed of light (C), this is the highest speed in this universe and this is the speed which can never be crossed.

Lorentz Transformation equations are the equations that deal with the phenomenon of “Moving frame” and “Rest frame” and also “time in moving frame” and “Time in rest frame”, we will share some interesting outcomes of Lorentz transformation equations.

 

Time Dilation: ----

      It is the theory of relativity which states that when a clock moves very fast then time starts to slow down in that clock.

 

Length contraction: ---

      From the relative theory of length contraction we can say that when an object moves very fast (velocity near  to 'C') and an observer is in rest parallel to the object, then the observer will notice that the moving object becomes shorter in length/ its length gets contracted.

Suppose there is a rod of 10 feet in a train and there are two observers, one is in the train and the other one is in earth, now if the train starts to move with a velocity near to ‘C’ then---

1) The observer who is in rest will measure the length of the rod as 8 ft, that’s shorter than original size. 

2)The observer in the moving train will measure it as 10 ft.



Now the question is----really it’s a length will contract if it travels with the speed of light?

Paradox: ----

    The term 'paradox' is from the Greek word 'Paradoxon' that means contrary to expectations existing belief of perceived opinion.

     It is a statement that appears to be self contradictory or silly but may include a latent truth. It consists of two meanings of the statement, both are true or both are false but meaning couldn't be true or false at the same time. It makes the reader to think deeply about it.

 

Twin paradox: ------

  It is a thought experiment in 'special relativity' in which identical twins, one of them travels in the space in a high speed ( appox 'C') rocket and then comes back to its original position to find his twin brother who was in rest in earth. It is likely to be happened that the travelled ones age will increase much slower than the rest one. We can prove it by the 'Lorentz transformation' rule and 'Relativity of time dilation'.

    Let us consider two twins Arun & Rahul. Both are 25 years old. Now, Arun makes a journey to the space in a rocket with a high velocity (approximate 'C', though it is not possible to apply this velocity in any object)suppose, It takes time 5 years to go to a destination and another 5 years to return…so total journey of 10 years, After coming back to its initial position, Arun will find that his age has become 35(25+5+5) but his brother Rahul’s age is 95 years.

This is really doubtful and complicated for all. When Rahul becomes 95 years old then Arun also must be 95 years old, but it didn't happen. Such thing happened because Arun traveled with the high velocity and time went slowed down to him which did not happen for Rahul as he was in rest in earth.

 

 

Ladder paradox: ----

 It is an experiment which requires a ladder, a garage and an observer.

 Now, we consider that the ladder is longer than the length of the diagonal of the garage and does not fit inside the garage, as we all know that the maximum length of the ladder that can fit in the garage is the diagonal of the garage.  Now, the ladder is moving with a very high velocity (approx 'C') towards the garage and the garage is in rest….so, according to Lorentz Transf. Equation the length of the ladder will be shorter than its original length, so the observer who is seeing the ladder from a rest frame will see that ladder’s length has contracted and then the ladder can be fitted in the garage----this one will happen really?

Now think about the opposite situation, there is a ladder of very short length, say one foot, and there is a garage of 10ft/12ft/12ft, now if the garage starts to move towards the ladder with a very high velocity (Nearly C) then according to L.T. equation the garage will be shorter and the ladder will fit in it------Puzzled?

 

Example of paradox: ------

(1) Your enemy's friend is your enemy: ---

   Let us consider A, B & C is three persons.

 A is B's enemy.

C is A's friend.

Now, you can say that C is also B's enemy. But it always never happens. We read "Rajam & Moni", where Rajam & Moni are enemy to each other but Swaminathan is friend of both Rajam & Moni. So we can say that your enemy's friend could be friend of you, never always enemy.

So here it creates a conflict between many people.

 

(2) You shouldn't go into the water, until you know how to swim: ----

      This sentence is very complicated and everyone gets confused in it and thinks deeply.

    If anyone doesn’t go into the water then how and where he will learn to swim??????

   And if anyone doesn’t know how to swim then he must not go into the water as there will be chance of drowning.

So you can see that it creates conflict.

But, the real meaning of these sentences are --- you can go into the water from where you can survive yourself, so at first, learn swim in a lower deep water and then go into the deep water.

 

(3) The beginning of the end: ----

       When a task is going to be end then, how it is in the beginning?

   But if the task was not started then it will not meet end. So, both are right. Every task has a start point and an end point, which are dependent on time. The beginning of the end means the task is in the beginning period of the end stage, which means starting of the end period.

 

 

(4) When a fan doesn't run, we can clearly see the blades and its joints with the head of the fan. But when a fan runs very fast, at that time we only see the head, the joint portion looks like vacant and creates a ring (blank), it seems that the wings create circle.

So, during running period someone who is in rest(we all) will think there are no wings ----but it’s not true, because the person who is sitting in the blade will see that different blades because he is moving with the blades. So, it creates conflict between two people.

 

 (5) Let us consider a train is running with a high velocity (approx 'c') and a rod & an observer(A) are present inside the train and an another observer(B) outside the train who is in rest position. After sometime B will notice that the train as well as the rod will contract but A will not notice any change because he was inside the train and traveled with the same velocity of the train.

 

 

 

 

(6) When a lorry is on a bridge and both length are equal. If the bridge moves very fast (velocity near to 'C') then what will happen?? Will the lorry fall from the bridge or not?? You will think that the lorry will fall from the bridge but it will not happen. As the lorry is on the bridge and the bridge is moving so the lorry is also moving with the same velocity of the bridge, then both of their lengths will contract. So, the lorry will not fall from the bridge---at least L.T. Equations says so.

So it can’t be said exactly whether Arun’s age will increase much slower than Rahul or not, it can’t be said exactly whether the ladder which is much longer than the diagonal of a room will fit in the room or not----until we make a journey with a speed of light.

But we should remember one thing---if we measure the speed a certain train standing in “earth” then it will not match the speed when measured from another “moving train”…so which sped is correct? Which frame (Earth, moving train) is proper?

 

Written by:-

Arnab(Math Honors ,2nd year)

Sanjiv(Diploma in Metallurgy, Final Year)