Active drag, part 3: two swims, one power. The perturbation family

The methods in this post are the ones most coaches will actually encounter poolside, because they need neither a flume nor an instrumented rod: only a pool, a timing system, and either an object of known resistance for the swimmer to tow or a dynamometer to tow the swimmer. Most of them share a single piece of algebra and all of them rest on one core assumption, and nearly everything worth knowing about them follows from those two things.

The algebra

Here $D$ denotes the drag that the perturbation model infers, and $P=D\cdot v$ its corresponding power; how these calculated quantities relate to the body’s frontal resistance and to its total external mechanical power depends on the physical interpretation explored in part 4. Suppose a swimmer produces the same power $P$ in two maximal swims. In the first, free swim she reaches speed $v_1$ against her own drag $D_1$; in the second she reaches speed $v_2$ against her own drag $D_2$ plus a known external force $F_{\mathrm{b}}$. Equating power across the two swims gives

$$P=D_1{v}_1=(D_2+F_{\mathrm{b}})v_2.$$

If drag scales as the square of speed, $D=k\cdot v^2$ with $k$ a constant for the swimmer, then $D_2=D_1\cdot {(v_2/v_1)}^2$, leaving a single unknown $k$ that can be solved. For a retarding perturbation ($v_2<v_1$), active drag at the free speed is

$$D_1=\frac{F_{\mathrm{b}}{v}_2{v}_1^2}{v_1^3-v_2^3},$$

and for an assisting one ($v_2>v_1$, with $F_{\mathrm{b}}$ pulling the swimmer forwards) the denominator becomes $v_2^3-v_1^3$. The power spent against drag follows as $D_1\cdot v_1$ (Kolmogorov’s school calls it total mechanical power; part 4), and a dimensionless drag coefficient as $D_1$ divided by $\frac{1}{2}\rho \cdot A\cdot v_1^2$, where $\rho$ is the density of water and $A$ whatever reference area the author prefers.

That is the whole method. Everything else comes down to making the two swims genuinely comparable and measuring the two speeds precisely enough.

The small perturbation method

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An animation of two maximal swims by one swimmer, shown in two lanes at once. A map of the 50 m pool marks 15 and 45 m. Lane one, free; lane two, towing a small water-filled cylinder under a float on a cord 3.5 to 4.5 body lengths long (drawn shorter, the cord broken, the body larger than it is). The cord runs from a belt on the swimmer’s back up over a small block sliding on a cable above the water, and down to the nose of the float, a long foam hull with the cylinder lying across its line of travel underneath. Each lane’s stopwatch runs from 15 to 45 m: the free swim takes 16.40 s, the towing one 17.23 s, about 1.5 m behind at the end. Below, the sums fill in as the numbers come: the free speed v1 = 30 / 16.40 = 1.8293 m/s; the speed with the load v2 = 30 / 17.23 = 1.7411 m/s; the speed loss 1 − v2/v1 = 4.8 per cent; the body’s calibrated drag at v2, Fb = 12.5 N; the same power in both swims, D1·v1 = (D2 + Fb)·v2, where D1 and D2 are the swimmer’s own drag in the free and the loaded swim, with D = k·v² and k a constant; so D1 = Fb·v2·v1² / (v1³ − v2³) = 12.5·1.7411·1.8293² / (1.8293³ − 1.7411³) = 86 N, the active drag at the free speed. The speeds are kept to four decimals: rounded to two, the same sum gives 85 N.

Kolmogorov’s group calls it the small perturbation method (SPM), first published in English by Kolmogorov and Duplishcheva (1992). The wider English-language literature came to know it as the velocity perturbation method, a phrase Kolmogorov and colleagues had used themselves in 1997 (Kolmogorov et al., 1997), though Kolmogorov prefers the original name (Kolmogorov, 2025); the abbreviation VPM spread after Toussaint et al. (2004). The swimmer performs two maximal swims over a timed 30 m section. In the second she tows a hydrodynamic body, a small water-filled cylinder under a float, on a line 3.5 to 4.5 body lengths behind, whose drag has been calibrated as a function of speed. A set of cylinders of different sizes allows the added resistance to be matched to the swimmer.

Published guidance on the size of the perturbation has shifted over time. The 1992 paper allowed a speed reduction of up to about 10 per cent (Kolmogorov & Duplishcheva, 1992), whereas Kolmogorov et al. (2021) and Kolmogorov (2023) explicitly restrict the speed loss $\mathrm{\Delta }{v}_0=(1-v_2/v_1)\times 100\%$, with $v_1$ the free and $v_2$ the loaded speed, to 4 to 6 per cent; the book (Kolmogorov, 2025) presents this narrower band as the optimum, and the 2021 paper calls it “the key point of the VPM (affecting the measurement accuracy)” (Kolmogorov et al., 2021, p. 4). Kolmogorov (2023) and the book go further, treating the 10 to 18 per cent used by several other groups as a violation of the method that systematically underestimates drag. Conversely, below 4 per cent, he writes, telling the two speeds apart would demand “devices of essentially higher accuracy” (Kolmogorov, 2023, p. 2): the metrological trade-off examined below, in his own words. I have not located the metrological study underlying this narrower interval in the material reviewed: the book refers to “special metrological studies” (Kolmogorov, 2025, p. 53; my translation) and attributes the criterion to Kolmogorov et al. (1997), the 2023 paper to Kolmogorov (2008) and Kolmogorov et al. (2021), while the 1997 paper in the Journal of Applied Biomechanics and the 2008 article do not state it. As we shall see, the interval matters considerably in practice.

The original 1992 validation rested on two checks: towing a swimmer with and without the body on a gravitational device (73.25 against 74.16 N, “the original error was about 1%”; Kolmogorov & Duplishcheva, 1992, p. 315), and comparing two bodies of different size on 20 front-crawl swimmers (50.21 against 50.99 N). The error from intra-cycle speed fluctuation was estimated by mathematical modelling at 6 to 8 per cent.

Thirty years later Kolmogorov (2023) compared three variants of the protocol on twelve high-level front crawl swimmers (780 to 850 FINA points). Two used the hydrodynamic stand built in 2011 over an outer lane of a 50 m pool, which gives active drag from a single 50 m swim. The body hangs from a slider running along a cable 0.5 m above the water and is towed from the 3 m mark to the 23 m mark, where an electromagnetic lock releases it; antennas at the side of the pool read a small sensor on the swimmer, giving the loaded speed over 10 to 25 m and the free speed over 30 to 45 m, each crossing timed to 0.01 s; a strain gauge in the line measures the force, and the cylinder’s volume is chosen to keep the speed loss within 4 to 6 per cent. The second variant on the stand swapped the segments (free first, loaded second), and the third, on the following day, was the classic two-swim test. Group mean drag in the three tests was 106.9, 107.0 and 108.3 N. His error figures are the largest differences between variants for a single swimmer: 2.4 per cent between the two stand variants on the same day (for power; 2.3 per cent for drag) and 5.4 per cent between the classic test and the stand the day before (for the coefficient; 5.2 per cent for drag) (Kolmogorov, 2023, p. 6). The systematic bias of −2.1 to −2.9 per cent was derived by computer modelling from Froude, Reynolds and Strouhal similarity rather than measured. For reliability he reports Pearson’s $r$ (0.979 for drag on the same day, 0.926 between days) rather than an ICC. My recomputation from the rounded individual values of his Table 2 (a two-way, absolute-agreement ICC for single measurements; typical error as the standard deviation of the differences divided by $\sqrt{2}$, in per cent of the standard stand’s mean) gives an ICC of 0.98 and a typical error of 1.0 per cent within the day, and 0.91 and 1.9 per cent between days, though for two different set-ups in a fixed order. It is the closest thing to a between-day check of the method; I found no published estimate of same-protocol test-retest error for the classic two-swim test in the sources reviewed. Notably, in the repeats of Toussaint et al. (2004), one swimmer’s values were 33.0, 39.0 and 50.6 N in three tests, the highest at a lower free speed.

The results made the method famous. Testing 73 Soviet national-team swimmers across the four strokes, Kolmogorov and Duplishcheva (1992) reported a striking result: active drag in front crawl, backstroke and butterfly below passive drag “in most of our swimmers” (p. 318; by the ratio column of their Tables 1 to 6 it is below in 25 of 67 tests, 37 per cent, some swimmers appearing in more than one stroke), whereas in breaststroke it was always above. Men’s front crawl values ranged from 42 to 167 N, and the drag coefficient, normalised to body volume to the power 2/3, from 0.17 to 0.46 against 0.24 to 0.29 when towed passively. The 1997 study of 797 swimmers (Kolmogorov et al., 1997) found no difference in the active coefficient between men and women at the same absolute speed, ranked the strokes (front crawl lowest, backstroke and butterfly similar, breaststroke highest), and reported, for men’s front crawl at 1.5 m/s, just 28 N against 53 N on the MAD system at about the same speed in the Dutch data (van der Vaart et al., 1987), a twofold difference the authors attributed to the level and effort of the swimmers. To explain how active drag could fall below passive drag in 1992, the authors offered tentative explanations: intra-cycle speed oscillations, which a towing test with a cam showed to reduce drag considerably, a smaller body area exposed to the flow, and the arm recovery out of the water (Kolmogorov & Duplishcheva, 1992). They also concluded that the active coefficient depends far more on technique than on body shape, a view developed in the 1997 paper and the book (Kolmogorov, 2025; Kolmogorov et al., 1997). Part 4 traces how the same group later reinterpreted what the method measures.

Variants abound. Wang et al. (2007) replaced the towed body with dry friction: a hollow metal sleeve sliding along a 3 mm steel wire stretched 0.49 m above a 50 m lane, its resistance set by three spring-loaded bolts pressing steel balls against the wire. A strain-gauge transducer sampled at 105 Hz joined the sleeve to a 4 m soft line clipped to the swimmer’s belt. Six national-level swimmers swam 25 m, free and loaded, four times in each condition in front crawl and breaststroke, timed by video from 7.5 to 22.5 m. The resisting force turned out to be “variable, not a constant value as Kolmogorov and Duplishcheva (1992) had assumed” (p. 377): the line slackened and tightened in step with the rise and fall of the swimmer’s speed within the cycle. Even so, the equation was fed its mean. Active drag in front crawl came out at 48.6 to 105.9 N for the three men and 36.3 to 50.3 N for the three women (p. 375). Several Portuguese studies use 25 m swims timed over 13 m and normalise to a frontal area, which gives coefficients that cannot be compared directly with Kolmogorov’s (Barbosa et al., 2013; Marinho et al., 2010; Morais et al., 2023); the normalisation belongs to the paper, not the country, as Alves et al. (2024), from the same tradition, normalised to body volume to the power 2/3. Finally, Gonjo and Olstad (2022) used a robotic resistance device on the starting block, and theirs is the study that finally put rigorous numbers on the method’s repeatability.

Assisted and resisted towing

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An animation of assisted towing. A map of the 50 m pool shows the dynamometer on its force platform at the far end. A side view follows the swimmer, towed on a line from a waist belt to the dynamometer at 1.98 metres a second, 7 per cent over his maximal free speed of 1.85; an arrow at the belt shows the tow force. Below, the tow force is traced through the run: it swings between about 12 and 35 newtons, dipping as each arm pulls, when the swimmer pulls himself and less tow is needed; its mean is 23 N. The sums, with v1 the free speed (1.85), v2 the towed speed (1.98), Ft the mean tow force and D1 and D2 the swimmer’s own drag at v1 and v2: the same power as free, D1·v1 = (D2 − Ft)·v2 with D = k·v² and k a constant, so D1 = Ft·v2·v1² / (v2³ − v1³) = 23·1.98·1.85² / (1.98³ − 1.85³) = 109 N, the active drag at the free speed: the small perturbation method’s equation with the sign reversed.

The Australian Institute of Sport inverted the perturbation. A dynamometer on a force platform at the pool end tows the swimmer through a waist belt at a speed set 5 to 10 per cent above her maximum, and the assisting force is recorded (Alcock & Mason, 2007; Mason, Toussaint, et al., 2013). In the constant-speed version the tow can reach 550 N. The “fluctuating” version therefore limits the force, to the swimmer’s mean passive drag (Mason et al., 2011) or, in Hazrati’s studies, to between a quarter and a half of it (Hazrati et al., 2013; Hazrati et al., 2015) or between half and three quarters (Hazrati, 2016), so that the swimmer’s speed can still rise and fall within the cycle. The main attraction is a continuous force-time profile synchronised with video, which the AIS used for immediate technique feedback (Mason, Kolmogorov, et al., 2013; Mason et al., 2012; Mason, Toussaint, et al., 2013; Mason et al., 2014). A resisted variant (RTM), pulling the swimmer back with a small force (12 to 21 N at the 2013 workshop, Mason, Kolmogorov, et al., 2013, where 10 to 20 N was recommended, Mason, Toussaint, et al., 2013; 4 to 10 N in Hazrati’s later work, Hazrati, Mason, et al., 2014), was tested alongside, and the workshop recommended averaging the two.

Repeatability is high, and appears highest where the apparatus is most coercive, though the published figures are not all of one kind: ICC 0.99 for constant-speed towing (Formosa et al., 2010), 0.91 for towing 10 per cent faster than maximal speed (Formosa et al., 2012), and for fluctuating towing 0.80 to 0.85 within a day for single trials and 0.92 to 0.93 between days for means of five trials (Hazrati et al., 2013; Hazrati et al., 2016).

The values themselves are the highest in the perturbation family. Constant-speed towing of 18 national-team swimmers gave 228 ± 11 N for men at 1.89 m/s against a passive drag of 79 N, about three times as much (Formosa et al., 2010). Fluctuating towing gave 78 to 169 N for men across Hazrati’s studies (Hazrati, 2016). The same ten swimmers towed assisted and resisted gave means of 105 and 91 N, statistically indistinguishable (Hazrati, Mason, & Sinclair, 2014) yet correlated across swimmers at only $r=0.32$ (Hazrati, 2016; by the thesis’s own table of individual values the correlation is even negative, part 5), so that Hazrati’s thesis concludes that “the same active drag values cannot be measured by the two assisted and resisted methods” (p. 164).

Hazrati’s thesis also holds the only full uncertainty budget in the literature. Propagating errors through the equation for twelve swimmers, with an assumed 7.5 per cent departure from equal power, she found a total uncertainty of 38 ± 13 N on a mean of 113 N, of which fully 36 N came from the equal-power term, 6 to 7 per cent from each speed, 2 to 3 per cent from the belt force and about 5 per cent from the exponent (Hazrati, 2016). The direction of the bias is predictable: when towed faster than their own speed, swimmers may relax and let the line pull them, so that drag is overestimated (Hazrati inferred this from the altered stroke phases); when held back, power falls and drag is underestimated. Nor does the speed profile under tow match free swimming: the within-cycle range is 13 to 25 per cent under tow against 19 to 36 per cent free, with altered stroke phases (Hazrati, 2016; Hazrati et al., 2015).

Kolmogorov’s book (Kolmogorov, 2025) regards towing at an externally imposed speed as a violation of dynamic similarity by the Strouhal criterion (the ratio of stroke frequency times a characteristic length to speed changes when speed is imposed from outside), and reads the 2013 workshop data (Mason, Kolmogorov, et al., 2013), in which assisted towing exceeded the perturbation method in the group tested over 10 m but the two methods came out in a different order in the group tested over 20 m, as a refutation of the ATM. From the same data the AIS group drew a more cautious conclusion, that the methods “certainly are not generally in agreement with one another” (Mason, Kolmogorov, et al., 2013). The numbers are the same; the interpretations are not.

Webb et al. (2011) offered a third variant from naval architecture (NABA): one swimmer was towed at 5, 10 and 15 per cent above maximal speed with the stroke rate meant to rise in proportion, so that the advance ratio stayed constant (the swimmer did not quite achieve it), and drag was derived with a thrust-deduction model from ship propulsion that does not assume resistance scales as $v^2$. The result, 133.9 ± 3.6 N, closely matched the perturbation formula applied to the same assisted tows (131.4 ± 4.4 N), while its run-to-run spread in each towing condition was 2.6 to 5.1 times smaller (standard deviations of 1.5 to 3.0 N against 6.0 to 15.2 N). It has never been repeated on a group of swimmers.

The assumption, and what it costs

Everything in the perturbation family hangs on one premise: “the same power in both swims”. No study reviewed has measured power directly in both swims, so the evidence on how well the premise holds is indirect.

Toussaint et al. (2004) tested six elite swimmers with the MAD system and the perturbation method, arms only in both. At 1.64 m/s the perturbation method averaged 53.2 N against 66.9 N on the MAD, with individual values from 44 per cent of the MAD figure to 22 per cent above it. When the authors regressed the drag discrepancy on the power discrepancy between the free and loaded perturbation swims, both powers estimated from each swimmer’s MAD curve (power in the loaded swim was on average 13 W lower), that power difference explained 88 per cent of the variance. A sensitivity analysis showed that “a 15% difference in power leads to 30% error in drag” (p. 1660). They also estimated that assuming $n=2$ instead of the measured mean of 2.34 shifts the result by a further 10 per cent or so.

A second account of the same underestimation, already suggested by Toussaint et al. (2004), is developed at length in Kolmogorov’s book (Kolmogorov, 2025). An oversized towed body changes the kinematics of the propelling limbs; power is diverted from useful work against frontal resistance into accelerating water backwards and shedding eddies; propelling efficiency falls; and the equation is fed a loaded swim that no longer represents the free one, even if the swimmer’s total mechanical power is unchanged. The two accounts predict the same sign of error, and existing data cannot separate them.

The perturbation dilemma

Here the newest data are the most illuminating: they point to a trade-off that can be derived rather than merely argued.

Gonjo and Olstad (2022) had eight men and eight women swim 25 m sprints against five loads from a robotic device (17 to 101 N for men, 16 to 58 N for women), twice on different days, and computed drag by the perturbation equation at each load, with the maximal free speed estimated from each swimmer’s load-velocity profile. Calculated drag fell as the load rose: for men from 83.5 N at 17 N added to 50.8 N at 101 N added (on the second day not at every step), for women from 59.4 to 41.0 N. At the heaviest loads drag fell below published passive values, which the authors read as clear underestimation. Repeatability followed a very different line: it was poor at the lightest load (16 to 17 N: ICC 0.40 for men, typical error 23 per cent), best at 37 to 60 N added (ICC 0.94 to 0.97, typical error at most 4.4 per cent), and lower again at the men’s two heaviest loads (ICC 0.91 to 0.92, typical error 6 to 7.6 per cent). They recommended 37 to 60 N as the working range while explicitly warning that the values in that range “are likely underestimated” (Gonjo & Olstad, 2022, p. 5).

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Two panels. Left, measured (Gonjo and Olstad 2022): VPM active drag at maximal speed against the added resistive force, falling as the load rises, for men (maximal speed 1.83 m/s) and women (1.57 m/s), each point labelled with its speed loss computed from their tables. Typical error between days for the men: 23 per cent at 17 N, 3 to 4 per cent at 38 to 59 N, 6 to 8 per cent at 80 to 101 N.

Group	Added force, N	Active drag, N	Speed loss, %
Men	16.6	83.5	9.3
Men	37.9	73.3	18.0
Men	59.1	67.3	30.6
Men	80.2	57.7	43.7
Men	101.5	50.8	59.6
Women	16.1	59.4	10.2
Women	26.6	52.6	18.5
Women	37.1	45.7	28.0
Women	47.7	44.3	36.9
Women	58.2	41.0	46.5

Right, predicted from the VPM equation: a conditional sensitivity of the formula with Fb fixed, not a measured error of any stand; the random error of the active drag, per cent (one standard deviation), against the speed loss in the loaded swim, Δv0, 0 to 40 per cent, for independent errors ε in each measured mean speed of 0.2, 0.5 and 1.0 per cent (about 0.03, 0.075 and 0.15 s in a 15 s swim): error ≈ √2·ε·(1 + 2r³)/(1 − r³) with r = 1 − Δv0, about √2·ε/Δv0 when the loss is small, rising steeply below 10 per cent. Marked: Kolmogorov’s optimum, 4 to 6 per cent; Gonjo’s most repeatable loads; and Gonjo’s observed typical errors for the men (at the heavier loads, 44 and 60 per cent speed loss, off the axis: 6.1 and 7.6 per cent). For small δ = Δv0, the active drag Da = Fb·v2·v1²/(v1³ − v2³) ≈ Fb/(3δ), where Fb is the load’s drag, v1 the free and v2 the loaded speed.

Converting their loads to speed loss gives $\mathrm{\Delta }{v}_0$ of 9 to 60 per cent for men and 10 to 47 per cent for women. Not one of their conditions fell within Kolmogorov’s 4 to 6 per cent (Kolmogorov et al., 2021), and a load small enough to do so would have been about 7 to 15 N, below the lightest and least repeatable load they used. The theoretical optimum and the measured repeatability sit at opposite ends of the scale.

This is not a contradiction: it is a trade-off built into the equation itself. Writing the loaded speed as $v_2=v_1\cdot (1-\delta )$, where $\delta$ is the fractional speed loss, the formula becomes

$$D_1=\frac{F_{\mathrm{b}}(1-\delta )}{3\delta -3{\delta }^2+{\delta }^3}\approx \frac{F_{\mathrm{b}}}{3\delta }\text{for small }\delta .$$

Because the denominator is proportional to the speed loss $\delta$, a small perturbation inflates the formula’s sensitivity to measurement noise. If each of the two speeds carries an independent relative error $\epsilon$ (from timing, from the swimmer’s own variation, from estimating $v_1$ rather than measuring it), the relative error in drag is, to first order, $\sqrt{2}\cdot \epsilon \cdot (1+2r^3)/(1-r^3)$ with $r=v_2/v_1$, about $\sqrt{2}\cdot \epsilon /\delta$ for small $\delta$. This is a conditional sensitivity of the formula itself, not a measured error of any stand: it assumes equal, independent relative errors in the two speeds and a fixed $F_{\mathrm{b}}$. If $F_{\mathrm{b}}$ is read from the body’s calibration curve at $v_2$, an error in $v_2$ changes $F_{\mathrm{b}}$ as well, and a full budget needs that coupling, the uncertainty of the force transducer and any correlation between the errors. At $\delta =5$ per cent, a timing error of just 0.5 per cent (0.075 s over a 15 s swim) gives about 13 per cent error in drag; to keep the random error in drag near 5 per cent, both speeds must be known to about 0.2 per cent, roughly 0.03 s per swim. (Kolmogorov’s 5.4 per cent is the largest difference between two variants for one swimmer, not a standard error. The paper reports a timing accuracy of 0.01 s for crossing each mark; this does not specify the standard uncertainty of a segment time or the covariance between timing errors (Kolmogorov, 2023, pp. 3, 6).) At $\delta =18$ per cent the same 0.5 per cent timing error gives about 3.3 per cent, of the same order as the typical error Gonjo and Olstad (2022) observed at that speed loss (3.6 per cent for men, 4.6 per cent for women); the comparison is only an illustration, since their free speed came from a load-velocity profile and the errors of the two speeds may be correlated. At their heaviest loads timing error should matter even less, yet repeatability fell again, so other sources of noise take over there. A longer timed section would shrink the timing error in proportion, but only so far: a swimmer cannot hold maximal speed for long; the loaded swim, being longer, tires them more and so strains the equal-power assumption further (below); their own variation from swim to swim does not shrink with distance; and in a 50 m pool the clean section between the start and the turn is about 30 m. Precise timing (electronic gates or video) and averaging $n$ repeats, which cuts the random error roughly by a factor of $\sqrt{n}$, reduce it without these costs. In short, a small perturbation protects the equal-power assumption and exposes the result to timing error; a large one does the reverse.

Crucially, the same amplification applies to violations of the assumption itself. Suppose the loaded swim is made at a fraction $q$ of the free swim’s power, with $k$ unchanged. With $r=v_2/v_1$, the formula then returns $D\cdot (q-r^3)/(1-r^3)$ instead of $D$, a relative error of about $-(1-q)/(3\delta )$. A loaded swim at just 1 per cent less power gives drag 7 per cent too low at a 5 per cent speed loss, and only 2 per cent too low at 18 per cent. A small perturbation is chosen precisely to protect the equal-power assumption, yet it magnifies whatever is left of its violation. An uncertainty analysis of assisted towing 5 to 9 per cent above free speed confirms the scale: a 7.5 per cent difference in power between the two conditions would lead to “about 30% error in calculated drag” (Hazrati et al., 2018, p. 7), more than any measured quantity contributed (5 to 6 per cent from the free and the towing speed, 2 to 3 per cent from the belt force); at their speed changes the formula above gives an amplification of 4 to 7.

Three practical consequences follow.

1. Under this independent-error, single-pair model, a drag uncertainty of about 5 per cent at a 5 per cent speed loss requires a relative standard uncertainty of roughly 0.2 per cent in each speed. A protocol’s own uncertainty budget must also account for the force, correlations and repeated trials, and the timing uncertainty should be stated explicitly.
2. Gonjo’s recommendation and Kolmogorov’s (Gonjo & Olstad, 2022; Kolmogorov et al., 2021) answer two different questions. Gonjo optimised test-retest repeatability, whereas Kolmogorov chose a small load to keep the swimming task as close as possible to free swimming, which is meant to protect the equal-power assumption but does not by itself establish it. Without very precise speed measurement, the method cannot satisfy both at once.
3. The single-swim stand (Kolmogorov, 2023) reduces the variation between trials, and reversing the order of its loaded and free segments gave virtually identical group means in his sample (106.9 and 107.0 N). Because power was not measured independently, however, this agreement does not establish constant power along the 50 m or exclude a bias shared by both variants: both tow the same body under the same equal-power model, and the paper estimates the effect of the load on the stroke only by modelling.

A breaststroke study shows the dilemma in practice. Alves et al. (2024) timed 25 m swims by hand over 13 m while citing the 5.4 per cent of Kolmogorov (2023), found on elite front crawl swimmers at speed losses of 4.4 to 5.7 per cent with automatic timing. Their 15 swimmers, aged 12 to 26, had 250 to 559 FINA points, and the speed losses in their Table 2 run from 3.2 to 15.7 per cent. Tellingly, the three largest changes in a swimmer’s drag between two breathing patterns, 85, 52 and 50 N, each involve a swim with a speed loss below 4.5 per cent; there, an error of just 0.01 m/s in either speed, about 0.1 s over 13 m, moves the drag by a fifth to a third.

For a coach, the operational advice is short. Use the perturbation methods when you want a single number at maximal speed in any stroke; record the speed loss and the timing precision alongside every result; run several repeats; compare only within the same swimmer and the same protocol; and judge a change against the 95 per cent threshold for a difference between two tests, about 2.8 times the typical error of one test (the standard error of a difference is $\sqrt{2}$ times it; part 5), not against the spread within a single test. Part 4 turns to the methods that avoid the two-swim comparison altogether by tethering the swimmer to a force sensor.

The series

1. What active drag is and why it has to be inferred
2. Pads and masks: the MAD system and the energetic method
3. Two swims, one power: the perturbation family (this post)
4. Residual thrust and the hydrodynamic reaction
5. Why the methods disagree, and what a coach can use

References

1. Alcock, A., & Mason, B. (2007). Biomechanical analysis of active drag in swimming. In H.-J. Menzel & M. H. Chagas (Eds.), Proceedings of the XXV International Symposium on Biomechanics in Sports (pp. 212–215). https://ojs.ub.uni-konstanz.de/cpa/article/view/438
2. Alves, M., Barbosa, T. M., & Vilas-Boas, J. P. (2024). Exploring the impact of breathing patterns on active drag in breaststroke swimming. Journal of Biomechanics, 177, Article 112390. https://doi.org/10.1016/j.jbiomech.2024.112390
3. Barbosa, T. M., Costa, M. J., Morais, J. E., Morouço, P., Moreira, M., Garrido, N. D., Marinho, D. A., & Silva, A. J. (2013). Characterization of speed fluctuation and drag force in young swimmers: A gender comparison. Human Movement Science, 32(6), 1214–1225. https://doi.org/10.1016/j.humov.2012.07.009
4. Formosa, D. P., Mason, B. R., & Burkett, B. J. (2010). Measuring active drag within the different phases of front crawl swimming. In P.-L. Kjendlie, R. K. Stallman, & J. Cabri (Eds.), Biomechanics and medicine in swimming XI (pp. 82–84). Norwegian School of Sport Science.
5. Formosa, D. P., Toussaint, H. M., Mason, B. R., & Burkett, B. (2012). Comparative analysis of active drag using the MAD system and an assisted towing method in front crawl swimming. Journal of Applied Biomechanics, 28(6), 746–750. https://doi.org/10.1123/jab.28.6.746
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