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

The methods covered in this post are the ones a coach is most likely to encounter in practice, as they require neither a flume nor a submerged rod of pads. An ordinary pool is enough, paired with accurate timing and either a small load for the swimmer to tow or a winch to tow the swimmer. All of these approaches rest on one simple idea and one underlying assumption, and nearly everything worth knowing about them follows from those two premises.

The idea: two swims and a small extra load

Picture two cars fitted with equally powerful engines: a boxy van and a sleek sports car. Hitch the same light trailer to each, one that adds just a touch of aerodynamic drag. The van, already shouldering its way through a heavy wall of air, scarcely notices the extra burden and loses only a fraction of its speed. The sports car, which normally slices through the air with minimal resistance, slows down noticeably more: relative to its own low profile, that same trailer is a substantial addition. In short, how much a car slows down when towing the trailer reveals how much drag the car itself generates.

A swimmer is tested in precisely the same way. They swim a set distance twice at full effort: once freely, and once against a small additional load whose resistance has been calibrated in advance. Provided the swimmer produces the same mechanical power in both trials, they will travel slightly slower with the load, and the extent of that slowdown reveals their own active drag. One further rule completes the calculation: drag increases with the square of velocity (“twice the speed means four times the drag”).

The small perturbation method: a float in tow

This test was devised by the Russian scientist Sergei Kolmogorov, who, along with a colleague, published it in English in 1992 as the “small perturbation method”. Here, the “perturbation” is a minor addition to total resistance that nudges the swimmer only slightly off their top speed.

In a 50-metre pool, the swimmer completes two all-out sprints, timed across the middle 30 metres (from the 15-metre to the 45-metre mark). On the second swim, they tow a small, water-filled cylinder suspended beneath a float on a line measuring three and a half to four and a half body lengths. Russian coaches nicknamed this swimming with a little boat. The cylinder’s resistance has been mapped beforehand across every speed, and a set of cylinders of varying sizes allows the load to be tailored to the individual swimmer.

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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.

The strengths of this approach are immediately obvious. The test works across all four competitive strokes, can be run in any standard pool, and costs very little. Its readings fall close to passive drag; in fact, in the first large-scale study conducted on the Soviet national team, active drag in front crawl, backstroke and butterfly came out even below passive drag in 37 per cent of the trials. The authors cautiously attributed this to the swimmer’s speed rising and falling within each stroke, the body presenting a smaller frontal area to the oncoming flow, and the arms recovering above the water. Another of their conclusions is of particular interest to coaches: active drag depends far more on swimming technique than on body shape.

How heavy should the extra load be? Kolmogorov insists that it must slow the swimmer by only 4 to 6 per cent. Other researchers have used heavier loads that reduced swimming speed by 10 to 18 per cent or more, a practice Kolmogorov regards as a violation of the method that artificially depresses the calculated drag. Why this distinction matters so much will become clear shortly.

In 2011, Kolmogorov engineered a rig that captures everything within a single swim. A cable is stretched above an outside lane, carrying the load on a carriage that slides along it. For the first 23 metres, the swimmer tows the load; an electromagnet then releases it, allowing the swimmer to continue unimpeded while poolside sensors record both speeds. In a 2023 study of twelve high-level swimmers, three variations of the protocol (two on the overhead rig and the classic two-swim version) yielded virtually identical group averages: 106.9, 107.0 and 108.3 newtons.

A winch instead of a float

The Australian Institute of Sport turned the idea on its head. In their setup, the swimmer is clipped by a belt to a winch at the end of the pool, which tows them 5 to 10 per cent faster than they can swim on their own, while a sensor records how hard it has to pull. The mathematics remain the same, only reversed: now the external force assists the swimmer instead of holding them back. There is also an inverted version in which the winch slightly resists the swimmer.

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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 winch offers two major advantages. Its measurements are highly repeatable, and it records force continuously throughout the entire stroke: data the Australians showed to swimmers right on the pool deck to fine-tune their technique.

Yet its readings are the highest in this family of methods: when men were towed at a constant speed, their active drag came out at roughly three times their passive drag. One likely explanation is that a swimmer being towed faster than their natural top speed may inadvertently relax and yield to the rope, making drag appear larger than it truly is. Conversely, when the winch holds them back, a swimmer may unconsciously ease off, making drag appear smaller. When the same ten swimmers were tested using both protocols, the group averages came out close, but an individual swimmer’s score in one test said almost nothing about their score in the other.

The big assumption and its cost

This entire family of methods rests on a single assumption: that the swimmer produces the exact same mechanical power in both swims. Because no study has measured that power directly across both trials, the validity of this assumption can only be judged indirectly.

Dutch researchers tested six high-level swimmers on both the MAD pads (from the second post) and the two-swim method, isolating the arms in both protocols. The two-swim method averaged 53 newtons compared to 67 newtons on the pads. Nearly all of that discrepancy (88 per cent) turned out to be explained by a single factor: against the extra load, the swimmers simply produced less power. By the researchers’ reckoning, if power output in the second swim drops by 15 per cent, the resulting drag calculation is skewed by 30 per cent.

There is a second explanation, which Kolmogorov lays out in detail in his book. Too heavy a load alters the mechanics of the arms and legs themselves. Part of the swimmer’s effort no longer goes into driving the body forward, but rather into throwing water backwards and churning up eddies. Even if total power output remains unchanged, the loaded swim ceases to resemble free swimming. Both explanations push the error in the same direction, and existing data cannot tell us which of the two is closer to the truth.

The trap: a small load or a big one?

Herein lies the method’s central dilemma, and it stems directly from the underlying maths.

A small load barely alters your stroke, so the equal-power assumption holds firm. However, the two swims then end up almost identical in speed, meaning drag must be calculated from a minuscule difference between them. It is like weighing a cat on bathroom scales by stepping on with the cat in your arms, then stepping on alone, and subtracting one number from the other. If the scales are off by half a kilogram and the cat weighs four, the relative error in the cat’s weight is fairly large: 12.5 per cent. For a small, skinny cat of 2.5 kg it grows to 20 per cent, and for a well-fed 5 kg cat it falls to 10 per cent. The same applies to the swimmer: if the load slows them by 5 per cent, a timing error of just 0.075 seconds over a 15-second swim throws the drag calculation off by about 13 per cent. To keep the error under 5 per cent, each swim must be timed to within roughly three hundredths of a second. Would a longer stretch help? It would shrink the stopwatch’s share of the error: twice the time, half the error. But nobody holds top speed for long; the swim with the float lasts longer and tires the swimmer more, so the two swims drift further apart in effort; the swimmer’s own variation from swim to swim does not shrink; and a 50-metre pool leaves only about 30 metres clear of the start and the turn. Better timing, with electronic gates or video, and several repeats averaged help far more.

A big load takes the edge off the timing problem: the gap between the two swims is wide, so a minor stopwatch error no longer matters as much. But a heavy load alters both stroke mechanics and power output. It is like testing a sprinter’s speed with a large parachute strapped to their back: they simply stop running the way they normally do. Consequently, the calculated drag comes out artificially low.

In 2022, researchers quantified this exact trap. Sixteen swimmers were tested against five different loads, repeating the entire protocol on a separate day. The heavier the load, the lower the calculated drag, and under the heaviest loads active drag fell even below passive drag, a clear sign that it was underestimated. Day-to-day repeatability was worst with the lightest load and best with loads that slowed swimmers by about a fifth, or even nearly a half: far beyond what Kolmogorov permits. The authors themselves candidly warned that within this most repeatable range the drag figures were likely too low. Not a single one of their test loads fell within Kolmogorov’s strict 4 to 6 per cent window.

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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.

In short, a small load protects the equal-power assumption but leaves the result at the mercy of the stopwatch; a big load does the exact opposite. Worse still, a small load also magnifies any residual breach of the assumption: if a swimmer puts out just 1 per cent less power during the loaded trial, a 5 per cent slowdown makes the calculated drag come out about 7 per cent too low.

A 2024 study of breaststrokers illustrates what this trap looks like in practice. The authors timed swimmers by hand over a 13-metre stretch, while citing the accuracy Kolmogorov had achieved with elite freestylers using automatic timing and a strict 4 to 6 per cent slowdown. In their own experiment, however, swimmers slowed by anywhere from 3 to 16 per cent under load. Tellingly, the three largest shifts in an individual swimmer’s calculated drag all came from trials where the swimmer slowed by less than 4.5 per cent: a zone where a manual timing error of just a tenth of a second over 13 metres swings the drag result by a fifth or even a third.

What to remember

The two-swim test is a practical tool when you want a single drag figure at top speed, in any stroke. However, it must be handled with strict discipline:

• Log the specifics. Always record the exact percentage of slowdown in the loaded swim and the precision of your timing system.
• Average your trials. Repeat the test several times.
• Keep comparisons consistent. Compare a swimmer only against themselves, and only using the exact same protocol.
• Respect the margin of error. Before concluding that a swimmer’s drag has genuinely changed, ensure the shift is nearly three times larger than the typical error of a single test, because each of the two tests being compared carries its own margin of error.

The next post looks at methods that dispense with comparing two swims altogether: instead, the swimmer is tethered directly 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

The sources for this post are listed in the full version.