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A group of swimmers have a training schedule - design the best way for them to do this.

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Maths Coursework        Newtons Laws        “Getting in the Swim.”


The Problem:

A group of swimmers have a training schedule. They must dive into the pool and swim one length of the pool. They must repeat this 20 times. They have been allowed one lane of the pool, so for obvious safety reasons they must all swim in the same direction in single file. My task is to find the bast way for them to do this. There are two possible models for this problem:

  1. Always swim in the same direction, climbing out after swimming a length and walking round the pool to where they started from and rejoining the queue.
  1. Climb out at the end of each length, wait for all the other swimmers to complete the length and then do the same in the opposite direction.


  • To simplify the model, I am going to assume that all the swimmers will swim at the same speed. For this speed, I have chosen 2ms-1, based on the speeds of some of my classmates.
  • I have chosen 1ms-1 as the walking speed, because it is slippery on a poolside, so it is unsafe to walk any faster.
  • I have also assumed that the swimmers won’t get tired, because this would obviously affect the swimming speed after 20 lengths.
  • I have decided to use a 50m pool (Olympic sized), and for model 1, I am assuming that they are using the end lane, and will only have to walk the length of the pool, not counting any distance walked along the width of the pool.
  • The swimmers all swim 20 lengths (and therefore walk 19).
  • There is a safety gap of 5 seconds. One-person jumps in and the next person waits 5 seconds before jumping in so that they don’t bump into each other. As the swimmers are all swimming at the same speed, their safety gaps will always stay the same.
...read more.


Model 1.

Always swim in the same direction, climbing out after swimming a length and walking round the pool to where they started from and rejoining the queue.

t = waiting time + time to swim one length + waiting time[1]

  = 5(n-1)+50+50

  1. 1

T = 20[5(n-1)+50]+(19*50)]


   = 100n+1350

To find out how many swimmers can be using the lane before they have to queue after every length I have drawn a timing diagram below. This shows that this model is fine for up to 15 people, because after the 15th person, there are still swimmers queuing for the previous length when the first swimmer has walked back.

      = 5 seconds (safety gap).image00.png

1 length = 75 seconds.

Swimmer                  A   B   C   D   E    F   G  H    I     J    K   L  M  N  O

Time (seconds)        0     5  10  15  20  25  30  35  40  45  50  55  60 65 70  75   80

After 15 swimmers, the extension to this model comes in. It is basically the same model, but with the queuing time added on.

Model 1 extension.

For more than 15 people, once the swimmer has swum a length, they will then have to queue again after walking back, because not every one will have started their previous length.

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        There were many simplifying assumptions made for this investigation (see introduction), and therefore this is only an approximate number of people. I think it gives a good idea of the numbers of people each model will cater for, even though it is not precise.

        To refine the experiment, I would have to take into account tiredness and the distance walked across the width of the pool. I would also have to take into account the speed of each individual swimmer. This would make the investigation much more limited as to the situation it could be easily used for, because if the speed of each individual swimmer is used, these will have to be found and integrated into the equation, which would make it very complicated and too time consuming to be effective. This way, it gives a rough estimate of which model a training scheme can use without being too precise to adapt to different swimmers taking part in the training schedule. In their present states, these models are only rough estimates (under estimates).

        As an extension, I could investigate the effect of different numbers of lanes on the investigation, and change the speeds of the swimmers, so that the models could be used for swimmers of various abilities.

P        05/08/07        Vicky Thornber


...read more.

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