Hypothesis:
I think that both models will produce an optimum time, but for a different number of people. Also, I think I will have to make a third model for an extension for model one, which is model one plus queuing time. After a certain number of swimmers there will be a queue at one end before the first swimmer can start their second length.
NB: n= number of swimmers
s = speed of swimmers
w = walking speed
t = time for 1 length
T= time for 20 lengths.
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
= 5(n-1)+50+50
- 1
T = 20[5(n-1)+50]+(19*50)]
2
= 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).
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. This will add time onto model 1, because for one length it takes 75 seconds, and then the second queuing time will have to be added on afterwards.
For more than 15 people, the time for all to swim 20 lengths increases by 5 seconds each time (the safety gap), so the new equation is:
n =15 T = 20[5(n-1)+50]+(19*50)]
2
n =16 T = 20[5(n-1)+50]+(19*50)] +19(5(16-15))
2
n=17 T = 20[5(n-1)+50]+(19*50)] + 19(5(17-15))
2
n>15 T = 20[5(n-1)+50]+(19*50)] + 19(5(n-15))
2
= 195n-75
Model 2.
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.
T=20(5n-5+25)+19(5n-5+25)
=195n+780
Analysis.
On the next page is a graph comparing the times of the three models for various numbers of people.. Under 15 people, model 1ext. is irrelevant, because model 1 is still correct. After 15 people, after every length the swimmers have to queue again. After 15 people, the model 1 is irrelevant , because it doesn’t take into account the queue that forms after 15 swimmers. As you can see, model 1 and model 2 cross at 6 swimmers. For 6 swimmers and under, model 2 appears to be better, but after that, model 1 and model 1 ext. are quicker. Therefore, model 2 is best for up to six people, and model 1 and therefore model 1 ext. are better for 6+ swimmers.
Conclusion.
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
