A Paddler Challenged Our Math (He Was Half Right)

A while ago we published a post arguing that clubs should rank time-trial results by meters per kilogram instead of raw meters. Divide the distance by the paddler's weight, the logic went, because the boat has to carry every kilogram it's given.

Then a paddler pushed back, and his argument was good. Heavier paddlers, he said, already pay for their weight in drag: more mass pushes the hull deeper, a deeper hull drags more, so raw meters are self-correcting and dividing by kilograms punishes big paddlers twice. He even had a neat per-seat comparison: an OC1 paddler moves their own weight plus about 12 kg of boat, and a dragon boat paddler moves their own weight plus roughly a twentieth of the boat, which is about the same number. In his view that also makes the OC1 the more honest solo time trial.

He was half right. So were we. The physics is worth walking through, because the correct answer is genuinely useful to any club that ranks its paddlers.

What drag actually does with a kilogram

Hull drag doesn't grow one-for-one with weight. Drag scales roughly with the wetted surface of the hull, and wetted surface grows with displacement to the power of two thirds. A paddler 20 percent heavier does not create 20 percent more drag in a solo boat: more like 12 or 13 percent.

Meanwhile, power grows roughly in proportion to muscle mass, so heavier athletes produce more absolute power. Put the two together for a solo craft (speed scales with the cube root of power over drag) and you get speed proportional to mass to the power of about one ninth. Tiny exponent, but positive: raw meters slightly favor the heavier paddler. Drag eats most of the weight advantage, exactly as our critic said. It just doesn't reverse it.

Now look at what the two scoring options do with that:

• Raw meters: slight advantage to heavy paddlers, roughly mass to the power 0.1.
• Meters divided by kilograms: massive advantage to light paddlers, roughly mass to the power of minus 0.9.

Dividing by kilograms overcorrects by about a factor of ten in exponent terms. A 95 kg paddler and a 55 kg paddler doing physiologically identical work will never look comparable on meters per kilogram. The heavier one basically cannot win, no matter how good they are. This is why indoor rowing's standard weight adjustment uses weight to the power 0.222, not weight to the power 1. The right correction exists, and it is small.

The dragon boat wrinkle

His OC1 comparison holds up per seat, but the drag logic breaks in a crew boat. Adding 10 kg to a fully loaded dragon boat, which weighs a couple of tonnes with crew, changes total drag by a fraction of a percent. In a dragon boat, extra body weight is nearly free and absolute power is nearly everything, so a strong heavy paddler almost always makes the boat faster. In an OC1, weight partially pays for itself, so the solo craft really is the more honest scale-free test. He was right about that, and it makes linear meters per kilogram even less appropriate for dragon boat data, not more.

Three questions, three numbers

The real lesson is that our original post blended three different questions into one number. They deserve separate answers:

• Who makes this boat faster? For a dragon boat: raw output, because the hull barely notices the kilograms. Weight still matters for trim and side balance, as the original post said, but as a seating problem, not a selection penalty.
• Who is the better athlete, pound for pound? Weight-adjusted meters: divide by weight to the power of about 0.22, the same family of correction rowing has used for decades. Small exponent, big fairness improvement, and a 95 kg paddler and a 55 kg paddler can finally meet on one leaderboard without the formula pre-deciding the winner.
• Who is improving? Neither of the above. Rank paddlers against their own recent baseline, say their trailing four weeks. Every paddler of every size can top that board, which is where the motivation actually lives, and cross-weight fairness debates evaporate when the competition is you against you.

What survives from our original post is its real point: the strongest paddler on the dock is not automatically the one moving the boat most, and quiet mid-sized paddlers are routinely undervalued. What changes is the formula. Linear division by kilograms is the one option the physics clearly rejects.

What we do about it

In practice, for a club: keep raw meters visible always, show a weight-adjusted number next to it and label it as adjusted, and run your motivational leaderboards on self-improvement rather than cross-athlete comparison. That's the setup we build toward in PaddleLens time trials too, because a ranking paddlers don't trust is worse than no ranking at all.

And if one of your paddlers argues with your scoring formula: listen. Ours was half right, and the half he was right about made the whole system better.