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Getting in the swim IntroductionA 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. 5 secs 25m ResultsThese are my first set of results.

These are the variables ? Length of pool? 25m Speed of swimming? 1msI 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. Person No Length?? 1 Length 2 Length 3 Length 4 Length 5 Length 6 Length 7 Length 8 Length 9 Length 10 1 25 95 165 235 305 375 445 515 585 655 2 30 100 170 240 310 380 450 520 590 660 3 35 105 175 245 315 385 455 525 595 665 4 40 110 180 250 320 390 460 530 600 670 5 45 115 185 255 325 395 465 535 605 675 6 50 120 190 260 330 400 470 540 610 680 7 55 125 195 265 335 405 475 545 615 685 8 60 130 200 270 340 410 480 550 620 690 9 65 135 205 275 345 415 485 555 625 695 10 70 140 210 280 250 420 490 560 630 700 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. Person No Length 11 12 13 14 15 16 17 18 19 20 1 725 795 865 935 1005 1075 1145 1215 1285 1355 2 730 800 870 940 1010 1080 1150 1220 1290 1360 3 735 805 875 945 1015 1085 1155 1225 1295 1365 4 740 810 880 950 1020 1090 1160 1230 1300 1370 5 745 815 885 955 1025 1095 1165 1235 1305 1375 6 750 820 890 960 1030 1100 1170 1240 1310 1380 7 755 825 895 965 1035 1105 1175 1245 1315 1385 8 760 830 900 970 1040 1110 1180 1250 1320 1390 9 765 835 905 975 1045 1115 1185 1255 1325 1395 10 770 840 910 980 1050 1120 1190 1260 1330 1400 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. Swimmer no Length 1 2 3 4 5 6 7 8 9 10 1 12.5 70 127.5 185 242.5 300 357.5 415 472.5 530 2 17.5 75 132.5 190 247.5 305 362.5 420 477.5 535 3 22.5 80 137.5 195 252.5 310 367.5 425 482.5 540 4 27.5 85 142.5 200 257.5 315 372.5 430 487.5 545 5 32.5 90 147.5 205 262.5 320 377.5 435 492.5 550 6 37.5 95 152.5 210 267.5 325 382.5 440 497.5 555 7 42.5 100 157.5 215 272.5 330 387.5 445 502.5 560 8 47.5 105 162.5 220 277.5 335 392.5 450 507.5 565 9 52.5 110 167.5 225 282.5 340 397.5 455 512.5 570 10 57.5 115 172.5 230 287.5 345 402.5 460 517.5 575 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 Swimmer No Length 11 12 13 14 15 16 17 18 19 20 1 587.5 645 702.5 760 817.5 875 932.5 990 1047.5 1105 2 592.5 650 707.5 765 822.5 880 937.5 995 1052.5 1110 3 597.5 655 712.5 770 827.5 885 942.5 1000 1057.5 1115 4 602.5 660 717.5 775 832.5 890 947.5 1005 1062.5 1120 5 607.5 665 722.5 780 837.5 895 952.5 1010 1067.5 1125 6 612.5 670 727.5 785 842.5 900 957.5 1015 1072.5 1130 7 617.5 675 732.5 790 847.5 905 962.5 1020 1077.5 1135 8 622.5 680 737.5 795 852.5 910 967.5 1025 1082.5 1140 9 627.5 685 742.5 800 857.5 915 972.5 1030 1087.5 1145 10 632.5 690 747.5 805 862.5 920 977.5 1035 1092.5 1150 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. FormulaI 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

travellingL = 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????????? ??????????????????????????????????????????????????????????????????????? ExampleL = D/S??????????????????????????????????????????????????????????????????????????????????? L

= 25 / 1 = 25 T= (L

+ G (N-1)) * A????????????????????????????????????????????????? T

= (25 + 5(10-1)) * 20 = 1400The 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 = 1150This 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 ? 1ms30/2 = 15 (15+5(10-1))*20 = 1200If 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.ExtensionThe 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. ResultsFor 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 followsDistance 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. Swimmers at starting point Seconds gone Swimmers swimming or walking back 2-10 25 1 3-10 30 1-2 4-10 35 1-2-3 5-10 40 1-2-3-4 1,5-10 42.5 2-4 1,6-10 45 2-5 1,2,6-10 47.5 3-5 1,2,7-10 50 3-6 1-3,7-10 52.5 4-6 1-3,8-10 55 4-7 1-4,8-10 57.5 5-7 1-4,9-10 60 5-8 1-5,9-10 62.5 6-8 1-5,10 65 6-9 1-6,10 67.5 7-9 1-6 70 7-10 1-7 72.5 8-10 3-7 75 1-2,8-10 3-8 77.5 1-2,9-10 4-8 80 1-3,9-10 4-9 82.5 1-3,10 5-9 85 1-4,10 5-10 87.5 1-4 6-10 90 1-5 1,6-10 92.5 2-5 1,7-10 95 2-6 1-2,7-10 97.5 3-6 1-2,8-10 100 3-7 1-3,8-10 102.5 4-7 1-3,9-10 105 4-8 1-4,9-10 107.5 5-8 1-4,10 110 5-9 1-5,10 112.5 6-9 1-5 115 6-10 1-6 117.5 7-10 2-6 120 1,7-10 2-7 122.5 1,8-10 3-7 125 1-2,8-10 3-8 127.5 1-2,9-10 4-8 130 1-3,9-10 4-9 132.5 1-3,10 5-9 135 1-4,10 5-10 137.5 1-4 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.FormulaThese are the variablesWalking distance 25 + 5 + 5 = 35 Swimming speed = 1ms Walking speed = 2ms Gap between swimmers = 5 secs Amount of swimmers = 10T = 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 speedFirst length = D/S + W/WS = LOther lengths = G * NThere 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 lengthsT = D/S + W/WS + 9*G +? A-1(G*N)Eg??????? 25?? + 35/2????

+ 9*5 +?? 9(5*10) =???? 42.5 + 45 + 450 = 537.5So 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 biggerT = 50/1 + 40/1 + 9*10 + 9(10*10) =

T = ????????????????? ?????????=180 + (900) = 1080 secondsSo again the bigger the variables

are (except speed that has to be smaller) the bigger the total time????????? ConclusionsThe 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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