View Full Version : Help with a swmming math problem

Syd

August 24th, 2009, 01:22 AM

Mathematically, how do you work out 'three quarter pace"?

Let's say (for the sake of easy calculations) your best time for an event is 1 minute. In practice, you want to go three quarter pace. So a 100% effort will result in a time of 60 seconds. A 75% effort should result in a longer time. My rudimentary maths tells me I should divide 100 by 75 and multiply the result by 60. That works out to 80 seconds, which is a minute and 20 seconds. Why does that sound much slower than what I would normally envisage three quarter pace to be? Or is my maths completely messed up?

DeskJockeyJim

August 24th, 2009, 02:48 AM

You look on the money to me... .75*80 seconds=60 seconds if you run the math backward.

Do what I do when you interval seems long: be grateful:bolt:

As to whether that's what your coach envisions when (s)he says three quarter pace, I have absolutely no idea:D

Rykno

August 24th, 2009, 06:07 AM

I don't think you can just plugg in numbers and calculate effort in % of times.

(it wasn't as confusing in my head as it is in written form :)

if I swim (what I feel to be) 80% or 90% 100 free. maybe my 80% is 1:20 and my 90% is 1:10 but my 100% race time is 1:00.9

Using just math I should be 90% at 1:07, 80% at 1:16

so maybe the 100-90% works better than 90-80%, but after that it seems hard to measure effort and time in linear %, there should be some kind of sloping factor.

I hold 1:20 for most of my distance swimming meaning (roughly 76% of my 100 time), but that would make my fastest 400 time 4:04 and it's only 4:41 and my practice times around 5:10-5:20.

does that mean I swim my 400 times at 90% even though I feel like I am only swimming 80%??

qbrain

August 24th, 2009, 09:58 AM

Great question.

Let’s assume that your 100% effort is your best time in that distance relatively recently, and you are similar in shape and size as when you swam that time. So we know distance is constant, the body in motion is constant and Newton did the hard work.

Work = Force * Change in displacement. Change in displacement is constant since we are displacing a swimmer a fixed distance. Solve for Force.

Force = mass * acceleration. The swimmers mass is constant, solve for acceleration.

Acceleration is where we stop, or we get into calculus that should give us the same answer anyway. Acceleration is measured in distance / time squared units. Since distance is constant, this gives us our slope for time which is what we are interested in.

So decreased effort results in an exponential increase in time.

This is the formula I propose to calculate effort, based on the pseudo math above.

(1/E)^2 * B = T

E = effort (85% = .85)

B = best time (seconds)

T = decreased effort time (seconds)

Based on a 1:00 best time.

95% 1:06

90% 1:14

85% 1:23

80% 1:34

75% 1:46

70% 2:02

Plugging in my freestyle times, 95% is about as fast as I can hit in practice, 85% I can do a lot of repeats (maybe all practice long), and 80% is pretty slow recovery.

lefty

August 24th, 2009, 11:22 AM

Effort level should be calculated off heart rate not time. 100% effort for me is 12 x 100's on 1:20. 90% is on 1:25 and 75% is 1:30. At 1:20 I have a heart rate in the mid to upper 150's. at 1:30 it is in the low 120's.

qbrain

August 24th, 2009, 12:19 PM

Effort level should be calculated off heart rate not time. 100% effort for me is 12 x 100's on 1:20. 90% is on 1:25 and 75% is 1:30. At 1:20 I have a heart rate in the mid to upper 150's. at 1:30 it is in the low 120's.

The more I think about this, the more I think it is a bad idea.

If you max heart rate is 200 bpm, and that is your 100% marker, then a 95% effort if 190 bpm, right? So you maintain the same heart rate for a 50 or a 1500? I don't think that makes sense.

jim thornton

August 24th, 2009, 12:38 PM

Or is my maths completely messed up?

Maths first. Grammars second.

lefty

August 24th, 2009, 12:43 PM

The more I think about this, the more I think it is a bad idea.

If you max heart rate is 200 bpm, and that is your 100% marker, then a 95% effort if 190 bpm, right? So you maintain the same heart rate for a 50 or a 1500? I don't think that makes sense.

I should have been more specific: I was suggesting training off max aerobic heart rate, not max heart rate. That is why I gave the example set.

ande

August 24th, 2009, 12:47 PM

I don't precisely gauge percentages, I just guess the effort,

also the percent effort you swim at might not give you the same time differential

you might want to think about percent effort in terms of ranges

also your range expands or contracts depending upon

1) how many you're doing,

2) how much rest you're getting

3) how you're feeling that day Are you on or off? Feeling good or hurting?

4) how psyched you are, is a coach timing you, who's watching you, who's swimming beside you

5) what suit you're wearing

6) if you're starting from a push or from a dive

7) how rested are you? (tapered?)

8) how many swimmers in & beside your lane, (waves)

9) how far apart are you leaving? (drafting)

10) what pool you're in? &

11) set instructions (like 5 DK or breathe every 5 affect your performance)

here's a guess about my

scy range while in a B70, from a dive with decent rest

100% 48 - 51

095% 52 - 54

090% 55 - 58

085% 59 - 61

080% 62 - 65

as far as mathmatically figuring it out

take your time & multiply it by 1.25 for 25% slower

but I think about the effort I'm putting forth & try to be with in an acceptable range

Mathematically, how do you work out 'three quarter pace"?

Let's say (for the sake of easy calculations) your best time for an event is 1 minute. In practice, you want to go three quarter pace. So a 100% effort will result in a time of 60 seconds. A 75% effort should result in a longer time. My rudimentary maths tells me I should divide 100 by 75 and multiply the result by 60. That works out to 80 seconds, which is a minute and 20 seconds. Why does that sound much slower than what I would normally envisage three quarter pace to be? Or is my maths completely messed up?

pwb

August 24th, 2009, 12:49 PM

Maths first. Grammars second.

Jim, I think it is we Americans who are incorrect with respect to the use of "math" or "maths." Outside of the US, it is my experience that most people (both native and non-native English speakers) will refer to mathematics as maths, not math like we say. Given that we never say "mathematic," it seems to me that we Americans are either incorrect in saying "math" or just staying true to our passion for efficiency by dropping the "s."

I still say math, though, and have two degrees in the said subject.

With respect to the question, I'd go with qbrain's formulas if you're so inclined. As for me, I just go a little slower by perceived effort when the coach asks us to back it off to a lower % of effort. The exact pace varies from day to day depending upon what's going on in my life and training.

swimmj

August 24th, 2009, 12:50 PM

Maths first. Grammars second.

Oh, silly Jim. Syd is using British english. They do maths and go to hospital, instead of doing math and going to a hospital. And don't get me started on chips (french fries), crisps (potato chips), biscuits (cookies), etc.

That Guy

August 24th, 2009, 02:09 PM

The best definition I ever heard for determining 75% effort involves no calculation whatsoever. Whatever your best 100 time is, swim a 75 in that amount of time. :banana:

scassady

August 24th, 2009, 03:47 PM

Great topic for discussion

Perhaps we should consider the two separate issues:

- Physiological response to exercise stress (i.e. heart rate, repirotory rate, lactic acid build up etc.)

- Power resistance "curve" provided by swimming activity

The power resistance "curve" for swimming activity would be where you could find the answer to the original question of a 3/4 pace swim to simulate competition effort.

Hydrodyanamic drag of a swimmer body through the water increases as the velocity squared. As a result, the power required to make a interval time will increase as the square of the relative change in time. Consider the following equations

F_drag = C_flow * V^2

V_swim = d_swim/t_interval

W_swim = F_drag * D_pool

P_swim = W_swim * t_interval

where:

F_drag - hydrodyamic drag of swimmer body through water

V_swim - swimmer velocity through water

d_swim - pool distance swum

t_interval - competition time or training pace interval

W_swim - work expenditure of swimmer. This quantity is directly related to required caloric intake to recover from swim workout

P_swim - power expenditure of swimmer. This quantity is related to swimmer bodies capability to convert chemical energy stores into usefull muscular power output

I put in the term C_flow as a constant that would represent an individuals flow resistance in the water. It considers skin friction, form drag and stroke efficiency. You would determine your personal C_flow number from swim tests. You might have one C_flow for sloppy swimming and one for high quality ;)

If you take all these equations and combine you could get the following:

P_race = C_flow * (d_swim^2 / t_race)

So if you consider C_flow and d_swim the same between train and race conditions and wanted an interval that was three quarter (75%) of your race speed power output you would just have.

t_train = t_race/0.75

You were correct in your original answer.

For me the challenge that engages me in those long sets of training intervals is working toward keeping efficiency high (make C_flow low) for strokes and turns and to simulate short periods of race conditions in training to monitor any changes in C_flow.

Happy swimming.

ande

August 24th, 2009, 04:10 PM

the problem with your math is each 25 would take 1/3 of your 100's time

so if you expand that out to 100

isn't it

4 x 1/3 = 4/3 or 133 1/3%

which is 33% more

time wise if you go 1:00 in the 100 free

wouldn't 75% effort or 25% slower be

1:15

The best definition I ever heard for determining 75% effort involves no calculation whatsoever. Whatever your best 100 time is, swim a 75 in that amount of time. :banana:

orca1946

August 24th, 2009, 08:01 PM

OK, I'll ask. Why would you want to swim @ 1/4 pace ??

jim thornton

August 24th, 2009, 11:41 PM

He brought his maths home.

Maths is his favorite subject.

His maths is improving.

He buys his maths supplies from the local maths supply warehouse, Maths Is Us. A rival chain, Maths R Us, was so roundly ridiculed by Europeans that it had no choice but to shut down. The former manager is in hospital.

I used to have a head for maths, but of late maths is flying out my ears. Wait, that is impossible. Maths is flying out one of my ears.

If a maths is cut in half and each piece regenerates, are you left with two mathses?

And so wiles away the waning hours of a long day spent looking at the maths in my checkbook. Given the negative numbers featured therein, the plural implications of "maths" seem particular cruel to me now. But perhaps I am being overly sensitive.

Perhaps some warm milk and a biscuit and bed, that's the ticket.

That Guy

August 25th, 2009, 12:05 AM

the problem with your math is each 25 would take 1/3 of your 100's time

so if you expand that out to 100

isn't it

4 x 1/3 = 4/3 or 133 1/3%

which is 33% more

time wise if you go 1:00 in the 100 free

wouldn't 75% effort or 25% slower be

1:15

The obvious corollary here is that it is impossible for any two people to agree on what 75% effort actually means. I don't use % efforts in my workouts, but if I did, I'd use the definition I provided, because it is so simple, it can be applied even when the maths-challengeds are too tireds to calculates diddlys/squats.

Syd

August 25th, 2009, 01:49 AM

He brought his maths home.

Maths is his favorite subject.

His maths is improving.

He buys his maths supplies from the local maths supply warehouse, Maths Is Us. A rival chain, Maths R Us, was so roundly ridiculed by Europeans that it had no choice but to shut down. The former manager is in hospital.

I used to have a head for maths, but of late maths is flying out my ears. Wait, that is impossible. Maths is flying out one of my ears.

If a maths is cut in half and each piece regenerates, are you left with two mathses?

And so wiles away the waning hours of a long day spent looking at the maths in my checkbook. Given the negative numbers featured therein, the plural implications of "maths" seem particular cruel to me now. But perhaps I am being overly sensitive.

Perhaps some warm milk and a biscuit and bed, that's the ticket.

Ha! Ha! Very funny, Jim. Where I learnt my English, the subject is called Mathematics (One would think 'maths' would be the obvious short form, but apparently not). Indeed, the North American rendering of Math has always seemed strange to us. (But in deference to you all, you will notice I included 'math' in the thread title). I somehow slipped up when writing the thread. Apologies!

I was also taught that gas is actually petrol, colour is spelled with a 'u', the trunk of a car is a 'boot' and the hood a 'bonnet'. My teacher did say, however, that to refer to someone as 'a FRUIT CAKE', has nothing to do with their propensity for eating the stuff.

..the plural implications of "maths" seem particular cruel to me nowYou're particular funny! :bolt:

fondly,

Syd

Chris Stevenson

August 25th, 2009, 03:24 AM

I was also taught that gas is actually petrol, colour is spelled with a 'u', the trunk of a car is a 'boot' and the hood a 'bonnet'.

I went to a British school in my teens when I lived overseas, and was forced to use spellings like colour and defence. While I was happy to study ALL maths, not just the one, I have to admit that their tendency to refer to erasers as "rubbers" was a little confusing to me in these formative years. ("You want to borrow my WHAT? And why would you think I would want it back?") :bolt:

qbrain

August 25th, 2009, 11:47 AM

Great topic for discussion

Perhaps we should consider the two separate issues:

- Physiological response to exercise stress (i.e. heart rate, repirotory rate, lactic acid build up etc.)

- Power resistance "curve" provided by swimming activity

The power resistance "curve" for swimming activity would be where you could find the answer to the original question of a 3/4 pace swim to simulate competition effort.

Hydrodyanamic drag of a swimmer body through the water increases as the velocity squared. As a result, the power required to make a interval time will increase as the square of the relative change in time. Consider the following equations

F_drag = C_flow * V^2

V_swim = d_swim/t_interval

W_swim = F_drag * D_pool

P_swim = W_swim * t_interval

where:

F_drag - hydrodyamic drag of swimmer body through water

V_swim - swimmer velocity through water

d_swim - pool distance swum

t_interval - competition time or training pace interval

W_swim - work expenditure of swimmer. This quantity is directly related to required caloric intake to recover from swim workout

P_swim - power expenditure of swimmer. This quantity is related to swimmer bodies capability to convert chemical energy stores into usefull muscular power output

I put in the term C_flow as a constant that would represent an individuals flow resistance in the water. It considers skin friction, form drag and stroke efficiency. You would determine your personal C_flow number from swim tests. You might have one C_flow for sloppy swimming and one for high quality ;)

If you take all these equations and combine you could get the following:

P_race = C_flow * (d_swim^2 / t_race)

So if you consider C_flow and d_swim the same between train and race conditions and wanted an interval that was three quarter (75%) of your race speed power output you would just have.

t_train = t_race/0.75

You were correct in your original answer.

For me the challenge that engages me in those long sets of training intervals is working toward keeping efficiency high (make C_flow low) for strokes and turns and to simulate short periods of race conditions in training to monitor any changes in C_flow.

Happy swimming.

I believe there is a mistake in your math such that P_race should = C_flow * (d_swim^2 / t_race^2).

scassady

August 25th, 2009, 09:45 PM

Thanks for the review. I do not think there is a mistake when I expand out the terms:

P_swim = ((C_flow * V^2) * D_pool) * t_interval

keeping in mind V = D_pool / t_interval

let us now substitute terms for V

P_swim = ((C_flow * (D_pool/t_interval)^2) * D_pool) * t_interval

one of the t_interval's gets canceled out.

srcoyote

August 28th, 2009, 09:40 AM

We have completely failed to take into account the fluid dynamics of the situation. As effort increases, even the smoothest stroke results in turbulence which would have an effect on the time ratios for the same distance. :D

I guess I have always thought of 3/4 pace or 80% pace as an intuitive understanding based on general feeling and experience. 80% is my cruising speed under which I could swim with effort, but could last 5 or 6 K swimming at that pace. 90% is what I would pace myself in the non-sprinting portions of a 1000 or 500. I haven't seen a workout ask for anything less than this nebulous 3/4 or 80% pace.

qbrain

August 28th, 2009, 11:45 AM

Thanks for the review. I do not think there is a mistake when I expand out the terms:

P_swim = ((C_flow * V^2) * D_pool) * t_interval

keeping in mind V = D_pool / t_interval

let us now substitute terms for V

P_swim = ((C_flow * (D_pool/t_interval)^2) * D_pool) * t_interval

one of the t_interval's gets canceled out.

Sorry, I don't have a Mech E background so I had to look some things up.

Drag Force = (1/2*p*C_D*A)*v^2

Everything in the () can be treated as a constant, which you called C_flow.

Drag Force = F_drag

F_drag = C_flow*v^2

velocity = distance / time

v = d/t

work = force * displacement

We will assume that displacement and distance are the same

W = F*d

power = work / time (not work * time)

P = W/t

To swim at a constant speed, you only need to overcome the force of drag. To convince yourself that this is true, think of iceskating (or rollerskating). Once you get going, you can glide for a long time with no effort, until the friction of the ice slows you down. Acceleration is hard, but once you hit you get going, maintaining speed is easy.

W_drag = F_drag * d

W_drag = C_flow * v^2 * d

W_drag = C_flow * (d/t)^2 * d

W_drag = C_flow * d^2/t^2 * d

W_drag = C_flow * d^3/t^2

Since C_flow and d are constants, the work needed is inversely related to the time. Thus each second faster you want to go is twice as hard as the previous second.

Now if you want to look at the power needed

P_drag = W_drag / t

P_drag = (C_flow * d^3/t^2)/t

P_drag = C_flow * d^3/t^3

Again, ignore the constants (or loose weight and swim shorter distances). A cubic increase in power is needed to drop time.

I am confident that the relationship bewtween work (or power) and time is an inverse exponential. Since the original question was about how to calculate a time to swim X, I believe the exponent is 2 and any linear portion of the equation can be ignored because it is dominated.

jim thornton

August 29th, 2009, 08:15 PM

I am confident that ... any... portion .... can be ignored because it is dominated.

Via just a bit of condensing, Q, I have managed to reduce your argument to a more elemental, and elegant, state.

You're welcome!

geochuck

August 29th, 2009, 08:28 PM

The coach at University of Toronto used to use the term 3/4 Pace. What he wanted was a relaxed almost 100% speed.

Do you really think it means to swim 75 yards in the time you swim 100??? I don.t think fluid dynamics even comes into this.

SolarEnergy

September 1st, 2009, 10:07 AM

Has anyone here already tried or is thinking about trying the Swim Score model?

Currently I am fair guessing my swim intensity into wko+ but in few weeks when I really start to train seriously, I intend of logging actual power equivalent, calculated using Skiba's Swim Score

ref.: http://www.physfarm.com/swimscore.pdf

Kevin in MD

September 1st, 2009, 02:22 PM

I've used the sharp stress score for quantifying swim training load. Easier in practice. The easiest though, is to use the Sharp score of a swim and multiply by yards instead of minutes. Well technically the easiest is just to use yards, but that is subject to pretty big variations depending on the quality.

The method you use to quantify your training load is not nearly as important as what you do with the number though. Phil Skiba's software implements the Banister model and gives great info on training load, effectiveness and taper.

So yes, I have looked into it but not found the plusses to outweigh the minuses of entering the data into a spreadsheet for every workout. I use sharp stress score, which is actually implemented into personal swim manager autoamtically.

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