Skip to main content

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.
       
Labels:

Comments

Post a Comment

Popular posts from this blog

LISSAJOUS FIGURES

  Definition:  "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures".      We know sine wave motion and circular motion is basically same.  Hence we draw two circles A and B perpendicular to each other.  The circle B rotates twice faster than circle A.  That is, frequency of circle B is two times than that of A.        A particle at the intersection of two circles is subjected to two sine wave motion   A and B at 90 degree simultaneously.  The particle will describe figures depending on the frequency and phase of A and B .  In our case, the ratio of frequency is  1:2 and the two waves are in phase.        To draw lissajous figures :  A moving point in both the circles are chosen.   Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two.  Hence the points are marked on the circles according to their speed.  Then straight lines
Post a Comment
Read more

THE PARABOLA

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.      Inverted parabola shape is used in the construction of buildings and bridges.  Because the shape is able to bear more weight.      A plane
Post a Comment
Read more

CASINO'S GAME

           Let us find out how the casino survives with mathematics.      Say, your friend invite you for a game of dice.  You must bet (wager) 2 dollars.  If you roll 'six' you will get back 8 dollars.  The game will go on for 30 rounds.  All sounds good.      The probability of rolling 'six' is 1/6.  Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5.  That is, you are expected to win 5 rounds out of 30.  Hence your gain will be 5 * 8 =40 dollars.  ok.  This also implies that you will loose 25 rounds.  Hence your loss will be 25*2 =50 dollars.  Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars.  There will be a loss of -1/3 or 0.33 dollars per round.  It is not a fair game.     Let us make a simple formula to calculate  'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect
4 comments
Read more