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
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.
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.
