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THE MATHEMATICS THAT IS RUNNING BEHIND USAIN BOLT

   


  At present the world record for 100 m dash remains at 9.578 sec by USAIN BOLT.  Will it further come down?  Will ever a man make it under 9 sec.  We will try to answer these questions via mathematics.  

 Harmonic series:


     Take a deck of 52 playing cards.  Move first top card to half of its length.  That is the maximum distance you can move before gravity pulls it down.  Now consider top first and second card as one unit and move both of them to 1/4 of the card length.  Again both will stay without falling.  Now consider first 3 cards as one unit and move all of them to 1/6 of the card length.  All the 3 cards stay put and we get some length of "overhang". Repeat the process for all the 51 cards.(last card cannot be moved).  The overhang accumulates as given by the following series.

 = 1/2+1/4+1/6+1/8+.........+1/102
= 2.2594........card length   

      We get a overhang of more than 2 card's length.  This arrangement seems to defy gravity.  Now coming back to the series, 

 = 1/2+1/4+1/6+1/8+.........+1/102
= 1/2[1+1/2+1/3+1/4...........]
The series [1+1/2+1/3+1/4...........]  is called harmonic series.
= 1/2*harmonic series

       There is a formula to calculate the sum of the series for 'n' terms.

 sum = 1+1/2+1/3+1/4..........
       = ln(n)+0.58                                 'ln' is log to base e = 2.71...
       = ln(51)+0.58
       = 3.93+0.58
       = 4.51
overhang = 1/2*(4.51)
               =2.255

      we get the same overhang.  If we increase the number of cards from 52 to 100000, what will happen?  The overhang will be 12.09 card length.

      So what we learn from harmonic series?  Even if we add very little by little, the sum goes on increasing though slowly.  There is no limit for the sum.

 Sub 9 second:
     We will now see how the 100 m dash world record had changed overtime.

 YEAR       RECORD        CHANGE
1980           9.95                 0
1990           9.92                -0.03
2000           9.79                -0.13
2010           9.575              -0.212

   Hence the time decreased little by little.  But it is faster than harmonic series.
     Even if the record changes in "harmonic fashion", we will reach sub 9 second record in 123600 years.  In my opinion, it will be much quicker because the training methods, the man physiology ( Darwin's theory) and accuracy of time measurement will all improve.

      Let us verify our mathematical prediction- 123600 years.  Let us assume that a new record will be created every decade.  So what will happen in 12360 decades.

 The sum of 12360 terms in harmonic series
 = 1+1/2+1/3+1/4..........1/12360
= ln[12360]+0.58
= 9.42+0.58
=10
We have to divide the sum by 10 since we have considered decades.
=10/10 = 1 second

 Hence according to harmonic series, one second change will take place in 12360 decades.  
Mathematics says that no record is unbreakable in the long run.
       
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