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Swimming Problem Maths Investigation

Introduction

A group of swimmers are following a training schedule that requires them to dive into the water and swim one length of the swimming pool. They must keep doing this until they have completed 20 lengths. For safety’s sake they have been allocated a single lane of the pool and all the swimmers must swim in the same direction in single file.

Half of the swimmers say that it will be quickest always to swim in the same direction, climbing out of the pool at the end of each length to rejoin the queue. The other swimmers want to climb out of the pool at the end of each length, wait until all the simmers have completed the length and then swim back one by one in the opposite direction.

I have to pick the method that I prefer and I have chosen the one where they get out at the end of each length, wait until all the swimmers have completed the length and then swim back in the opposite direction. I have chose this way because then the swimmers don’t have to waste their energy walking from one side of the pool to the other all the time and they can use their energy to swim.

To set up a model for this experiment first of all I have to make a few assumptions.

First of all I have to think of a set distance for the length of the pool. Because this is a variable it will make a difference in the time that it takes to do one length of the pool.

 I also have to assume that all the swimmers are swimming at the same speed and at a constant speed all of the time.

Each person has to dive in the pool so many seconds after the last one, this I obviously or safety so people do not dive onto each other.

I will set up an appropriate model keeping in mind the assumptions that I have made.

Results

These are my first set of results. These are the variables –

Length of pool  25m

Speed of swimming  1ms

I have chosen these variables because I think that they are quite realistic values. 25m is about the size of most swimming pools and I would estimate that 1ms is a quite realistic speed for a swimmer to be swimming at.

This table shows the total time at each point during the twenty lengths the time is measured in seconds. For Example Swimmer four will have done 4 lengths 250 seconds after the swimming session started. The total time for ten people to complete 10 lengths is 700 seconds. I will now go on to do the same for the next ten lengths because the session is not over until they have done twenty lengths each.

There is a delay between each length because swimmer no 1 has to wait before swimmer 10 has finishes his last length before he can start his next length. This takes quite a lot of time while he/she is waiting to do his/her next length.

The total time it took for the swimmers to complete 20 lengths swimming at a constant speed of 1ms (metres per second) and starting swimming five seconds after the last person is 1400 seconds which is 23 minutes and 20 seconds.

I will now change the variables to see what happens to the results. I will keep the length of the pool at

25 metres but make it so the swimmers are swimming at a faster pace of 2ms. Here are the results.

I took 575 sec for 10 people to swim 10 lengths of a pool, which are 25 metres long and swimming at a constant rate of 2ms. This is a lot less time than when they were travelling a 1ms.

These is the results of the next 10 lengths

The total time it took for the swimmers to complete 20 lengths swimming at a constant speed of 2ms (metres per second) and starting swimming five seconds after the last person is – 1150 seconds which is 18 mins and 25 seconds.

Formula

I am now going to try and work out a formula in which to calculate the total time it takes to complete any amount of lengths at any speed and with any set distance between the swimmers.

The variables I am going to need for this formula are

T = Total time

L = Total time it takes for 1 swimmer to complete 1 length

A = Amount of lengths

N = Number of swimmers

G = Gap between two swimmers

D = distance of length

S = speed the swimmers are travelling

L = D/S                                                  

T = L + G*(N-1) * A                            

The formula that I have just worked out is a way of finding out the total time when any number of swimmers have completed and number of lengths at any speed and any distance and at any time apart from each other.

I will now check the formula with the results that I have found out already to check to see if it works.

Formula                                                                                                          Example

L = D/S                                                                                                   L = 25 / 1 = 25

T= (L + G (N-1)) * A                                                                            T = (25 + 5(10-1)) * 20 = 1400

The formula has worked because I have found the same answer to when I worked it out earlier. What I did was substituted all the variables into the equation and then worked out an answer.

I will now just double check the equation to make sure that it works for all numbers and not just for the last question.

L= D / S                                                                                                  L = 25 / 2 = 12.5

T = (L + G (N-1)) * A                                                                           T = (12.5 + 5(10-1)) * 20 = 1150

This has also given the same answer that I worked out earlier so I know that the formula works.

I will now change the length of the pool and use the formula to see what happens to the results.

Length of the pool – 30m

Swimming speed – 1ms

30/2 = 15

(15+5(10-1))*20 = 1200

If the distance of he length is greater then the total time it takes the swimmers to complete the lengths is greater.

So obviously the smaller the variables (except the speed because that has to be fast) you put into the equation the shorter the total time will be. If you enter high variables for everything except speed you will get a very high total time.

Extension

The swimmers have thought of another way to do their lengths and they want to know which way will be the quickest the method I have just examined or their new idea.

Their new idea is this.

Because the swimmers have only been allocated a single lane of the swimming pool so there is only a few ways that they could complete the lengths swimming in the same direction in single file for safety’s sake. The idea is that the swimmers complete the length and when they have finished climb out of the pool and walk to the other side of the pool and join the queue ready to complete the next length.

Here is a diagram of the path that the swimmers will take

This method will be different to the last one when I am making my calulations because I will have to add the walking time to the total time.

Results

For this model I have chosen to keep the distance of the length at 25m because it is a realistic value. I have used a swimming speed of 1ms like on the first model, because I think this is a realistic value to use. I have used 2ms for the walking speed because I think this is a realistic speed to be walking at.

The variables for this set of results are as follows

Distance of length 25m

Walking distance 25 + 5 + 5 = 35

Swimming speed = 1ms

Walking speed = 2ms

Gap between swimmers = 5 secs

Amount of swimmers = 10

All swimmers have done two lengths at this point

All swimmers have done one length at this point

I have decided to work out the first two lengths because the time taken to complete the first two lengths is different because on the first length it takes 25 seconds before the first one completes a length and starts walking back. On the second length he still does it in 25 seconds but its only five seconds after the last person finished so the length is quicker.

All swimmers have done one length at this point

All swimmers have done two lengths at this point

When I considering a formula for this method of doing the lengths I have to work out the time it takes to do the first length first, because this one takes more time to complete than all the other lengths.

Formula

These are the variables

Walking distance 25 + 5 + 5 = 35

Swimming speed = 1ms

Walking speed = 2ms

Gap between swimmers = 5 secs

Amount of swimmers = 10

T = Total time

L = Total time it takes for 1 swimmer to complete 1 length

A = Amount of lengths

N = Number of swimmers

G = Gap between two swimmers

D = distance of length

S = swimming speed

W = walking distance

WS= walking speed

First length =

D/S + W/WS = L

Other lengths =

G * N

There are 9 lengths other than the first one so I have to multiply this by 9 to get the last 9 lengths.

Here is the final formula for the full 10 lengths

T = D/S + W/WS + 9*G +  A-1(G*N)

Eg           25   + 35/2     + 9*5 +   9(5*10)

=     42.5 + 45 + 450 = 537.5

So the total time for the swimmers to complete the circuit of 20 lengths, swimming at 1ms and walking at 2ms is 537.5 secs. (8mins 57.5secs)

Here's what happens if I make the variables bigger

T = 50/1 + 40/1 + 9*10 + 9(10*10) =

T =                                   =180 + (900) = 1080 seconds

So again the bigger the variables are (except speed that has to be smaller) the bigger the total time  

Conclusions

The values of the variables make a difference in what the total time is, for all variables except speed the lower the value the lower the time. Speed makes the time lower when it is high, because obviously the quicker the swimmers are going the quicker they are going to finish.

In most occasions the second method, (where the swimmers walk from one side of the pool to the other) is the quickest, the only time it isn’t the quickest is when there is a very long length distance and a low number of swimmers. The problem was to find the quickest method over 20 lengths, so over 20 lengths the second method is the fastest.

The main reason for this is because with the first method, when nine of the swimmers have completed their lengths they have to wait for the tenth person to finish before the first swimmer can start his second length. With the second method there is always people in the pool it is continuous because they all go 5 seconds after each other so there is no waiting which puts the total time up.

I don’t think this is a particularly accurate set of models compared to life because I have assumed that all the swimmers swim at exactly the same speed, and they all walk at exactly the same speed, obviously this wouldn’t happen. There would be varying results for the time taken for each individual to complete a length.  Also swimmers would probably start to slow down each time, they get worn out and they will swim at different speeds through the 20 lengths. Swimmers might lose energy through walking the distance of one length so in real life the first method may be more appropriate. There is no real way of testing without very very accurate models based on exact results for the sets of swimmers.

If I had a lot more time and I was to change anything on this problem I would make the models more exact. I would find out actual swimming and walking times through testing and not just estimating. I would experiment on the average times it takes swimmers to swim 20 lengths. I would make the whole model more exact thus giving a more reliable result.

Based on the assumptions that I have made through this problem, the second method is the quickest way for the swimmers to complete 20 lengths. They walk using this method but they complete the 20 lengths in a lot less time than by using the other method.

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