What this is
An independent reimplementation of the athletics hammer throw — the World Athletics field event, not any existing computer game. The circle, the sector, the cage, the implement and every judging decision come from the World Athletics Book C – C2.1 Technical Rules (TR 25 competition format, TR 32 general throwing conditions, TR 36 the hammer throw and the hammer, TR 37 the cage), which was downloaded and read for this build. The thrower is a physics model, and the numbers it produces are compared below against measured championship data rather than tuned to match it.
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The release angle: derived, not assumed
A projectile thrown and caught at the same height goes furthest at 45°. Hammer throwers are widely said to release "well below" that. This build set out to derive the optimum from its own mechanics and then look at what throwers actually do. Two of those three statements turned out to need correcting.
What the model finds
| what is included | optimum release angle |
|---|---|
| a projectile from level ground | 45.00° |
| released 1.50 m up, landing at ground level, no air | 44.48° |
| … plus air resistance | 44.00° |
| … plus the turn mechanics, which make speed depend on angle | 41.55° |
So the coupling between speed and angle is real and it is the largest of the three corrections — 2.45°, against 0.52° for release height and 0.48° for air. Its mechanism in this model is geometric: the head's orbit lies in a plane tilted by β, and the release angle can never exceed β. Tilting the plane further shortens the radius the thrower can reach (the low point cannot go through the ground), and a shorter radius at the same speed means more wire tension — T = m v² sinβ / H on an orbit that grazes the ground. Past what the thrower can hold, the drive stops. Steeper costs speed.
What throwers actually do
The twelve men's finalists at the 2017 World Championships in London released at 41.28° ± 2.52° (range 36.7° to 46.2°); the twelve women at 40.65° ± 2.10°. This model's optimum, 41.55°, sits 0.27° from the men's measured mean, and it was never fitted to any release angle — the only things calibrated to that dataset were the thrower's drive torque, entry speed, orbit radius and plane height.
Three corrections
- "Well below 45°" overstates it. The measured men's mean is 3.7° below 45°, and the 2017 world champion released at 46.2° — above it. That is a correction to common framing, not to any source I read: the published sources are more careful than the folklore. The narrative review by Castaldi and colleagues (Frontiers in Sports and Active Living, 2022) says the optimum is "slightly < 45°", about 44° for throws of 60–80 m, and that athletes throw at 40–42°.
- The quoted "optimum" is the wrong optimum. Every published figure I found — 45° in the 2017 IAAF report's coach's commentary, about 44° in Otto's analysis of Sedykh's world record, "slightly < 45°" in Frontiers, and 44° used as a modelling assumption by Horváth and colleagues (2023) — is the ballistic optimum, which is below 45° only because the release is above the landing plane. None of them solves the trade-off Frontiers itself names. Linthorne did exactly that calculation for the shot put in 2001, where including the trade-off moved the optimum down by about 8°; his paper mentions the hammer in one aside and never applies the method to it. As far as I could find, no published work has solved the speed/angle trade-off for the hammer. The 41.55° above is this build's attempt, and it is a model result, not a measurement.
- The angle barely matters, and that is the real answer. At the measured release conditions the range surface changes by 0.57 m per degree of angle but 5.08 m per m/s of speed. One degree of angle is worth 0.113 m/s — four tenths of one per cent of release speed. Releasing at 37.6° instead of this model's optimum costs 0.67 m, which 0.47 % more release speed repays in full. In the 2017 men's field, distance correlates with release speed at r = 0.59 and with release angle at r = 0.05; in the women's field the speed correlation is r = 0.95. A thrower who trades angle for speed is not making a mistake.
Honest limit: this model's coupling is a factor 2.6 too weak to put the optimum as low as 37.6°. If a thrower really is best served releasing that flat, something couples speed to plane inclination more than twice as hard as anything here — and nothing I read measures it.
Two geometric predictions, and what the measurements say
The release angle can never exceed the plane inclination
The head runs on a circle in a tilted plane. The steepest its velocity can ever point is the plane's own inclination β, reached exactly a quarter turn past the low point; anywhere else the tangent is flatter. That is pure geometry and this engine derives it before it knows anything about throwing.
The 2017 reports publish both quantities, in different tables: the angle of release (Table 3) and the "relative upswing path angle", the inclination of the line from the low point to the high point (Table 11). Twenty-three of the twenty-four finalists satisfy α ≤ β. The single exception exceeds it by 0.2°, which is the tables' own reporting resolution. The mean deficit is 0.29°, so elite throwers release within a third of a degree of the steepest angle their plane allows — essentially at the top of the upswing. This model's optimum releases 8.7° earlier than that, trading angle for speed. That is a real disagreement between the model and the measurements, and the model is the thing more likely to be wrong.
A trap for anyone reading those reports: Table 3's "angle of release" and Table 11's "release" column are different quantities and differ by up to 1.0°. They are easy to conflate.
Where the low point has to be
If the throw must leave along the sector centre line, and the head is a quarter turn past its low point when it does, then the low point of the orbit must lie directly opposite the sector — at the back of the circle. The engine derives that from the aiming requirement alone; it is never told. The 2017 report measures the azimuth of every low and high point, with 0° at the back of the circle and 180° at the front. In the final turn the measured low points average 9.9° from the back and the high points 10.4° from the front.
A second consequence falls out of the same geometry: the horizontal ground track of the head is an ellipse, with semi-axes R cosβ and R, not a circle. Treating it as a circle gives the wrong azimuth everywhere except at exactly a quarter turn — an error this build made first and its own harness caught.
The thrower cannot stand still vertically
At the high point of a fast turn the wire pulls the thrower upward with a vertical component larger than the whole system weighs. For the ground reaction to stay non-negative the thrower's own centre of mass must be accelerating upward at that instant, which forces a vertical bob of at least 3.3 cm per turn, in antiphase with the head. Frontiers reports the measured phase lag between the thrower's and the hammer's centres of mass as about 115°, and notes that there are no specific studies of the thrower's or the system's centre of mass at all.
Where the published record does not close
These are the places where a document I opened turned out not to decide the answer, or not to agree with itself or with another document. Each says whose claim it is.
- TR 36 sets no minimum length for a hammer. The implement table gives a maximum length from the inside of the handle (1215 mm for the 7.260 kg and 6 kg hammers, 1200 mm for the 5 kg, 1195 mm for the 4 kg and 3 kg) and minimum and maximum head diameters — but no minimum length row and no manufacturers' weight-range row. The discus table in the same rulebook gives a minimum and a maximum for every dimension, and the javelin table gives both for overall length. The strings 1160, 1165, 1170 and 1175 do not appear anywhere in the 124-page document. A legal hammer may be arbitrarily short. Gap in World Athletics TR 36.
- Wikipedia states the hammer's length limit wrongly. Its article says the men's and women's hammers have "the wire in either case no more than 122 centimetres in length". TR 36 does not limit the wire's length at all; it limits the length of the whole implement measured from the inside of the handle, and the limit is not the same in either case — 1215 mm for the men's 7.260 kg hammer against 1195 mm for the women's 4 kg. Error in Wikipedia, corrected against the rulebook.
- TR 32's own sector-laying note is a chord, not a width. The rule says the sector may be set out by making the two points 20 m from the centre 12 m ± 0.05 m apart, "thus, for every 1 m from the centre of the circle, the distance across shall be increased by 0.60 m". Measured as a straight line between those two points, that is exact: 2 × 20 × sin 17.46° = 12.002 m, and 34.92° is 2 × arcsin 0.3 = 34.9152° rounded to two places. Measured as the perpendicular width of the sector it is not: 2 × 20 × tan 17.46° = 12.581 m, and the note read that way would imply a 33.40° sector, 1.5° too narrow. Ambiguity in TR 32; the chord reading is the one that closes.
- TR 25.17 leaves the throws' one-athlete time blank. The time-allowed table gives 1 min for more than three athletes, 1 min for two or three, 2 min for consecutive trials — and a dash in the "1 athlete" cell for every event outside the high jump and pole vault. Gap in World Athletics TR 25.17.
- The 2017 London release parameters only fly their own distances in a vacuum. Running the twelve published triples (speed, angle, height) through this build's flight integrator and comparing against the twelve published distances, the residual — which geometry says must lie between −1.07 m and about +0.4 m, because it is just where the release point sits relative to the circle — comes out at +2.59 ± 0.97 m on the published hammer drag, +1.34 m with the head's drag alone, and −0.85 m with no air at all. The scatter is small, so it is systematic: either the measured release speeds are about 1.7 % low, or the effective drag is far below the published value. Inconsistency inside the IAAF 2017 report, found here.
- Sedykh's published world-record parameters miss the other way. Otto's analysis (reprinted in New Studies in Athletics) gives 30.7 m/s at 39.9° from 1.66 m for the 86.74 m record. Those numbers are internally consistent — the published horizontal and vertical components resolve to exactly that speed and angle — but they fly 91.3 m, about 4.6 m further than the record, which no release geometry can absorb. Horváth and colleagues' reconstruction of the same throw, 29.68 m/s at 44° from 1.778 m, closes to within 0.03 m. The two measurement campaigns miss in opposite directions, so neither error is simply a wrong drag coefficient. Inconsistency in Otto / New Studies in Athletics, found here.
- Two published drag losses differ by nearly a factor of two. Frontiers (2022), citing Jánosi and Bántay (2002), says air costs about 3 % of the range of a 75 m throw. Horváth and colleagues (2023) publish k = 0.7 over Q = 0.0138 m² for the hammer, which in this integrator costs 4.8 %. Modelling only the head as a smooth sphere gives 3.0 %. Disagreement between two published sources.
- The published head-path radius depends on how you measure it. Murofushi and colleagues report 1.63–1.73 m from video (Gutiérrez-Dávila and colleagues, at the 1999 World Championships), 1.56–1.75 m from their own video, and 1.71–1.93 m from sensors mounted on the hammer itself — and say in terms that the difference is the method, not noise. Disagreement flagged by the source itself.
- Two analyses of the same athlete's tangential force disagree threefold. Susanka and colleagues (1987) report it fluctuating by ±500 N twice per turn in a 79.22 m throw by Sedykh; Bartonietz (1994) reports −150 N to +200 N once per turn in an 82.34 m throw by the same athlete. Disagreement flagged by Brice (2014).
- The Reynolds number at release sits inside the drag crisis. At 28 m/s a 130 mm head runs at Re ≈ 2.5 × 105, which for a smooth sphere is right in the transition where the drag coefficient falls from about 0.47 to about 0.20. That is exactly where a textbook number is least trustworthy, and it is part of why the measured-effective value published for the whole implement is worth more than a sphere curve. Derived here.
What this model does not do
This section is plain markup and does not depend on any script, because a disclosure that disappears when a script fails is not a disclosure.
- The centre of the head's orbit is held at a constant height. Real throwers lift it into the delivery: Fajdek's measured turn-4 low and high points imply an orbit centre near 1.30 m, while his measured release height was 1.69 m. A single rigid plane cannot match both, so this build matches the release height, which is what governs the flight.
- The measured final turn lengthens again (Berlin 2009: about 0.60, 0.50, 0.46 then 0.51 s) because of the delivery. This engine's turns only get shorter.
- The coefficient that makes a steeper plane cost drive — how much of the thrower's torque is bound to the vertical axis — is not measured anywhere I could find. It is set to 1 here, meaning all of it, and even that is a factor 2.6 too weak to explain the flattest measured releases.
- A hammer that strikes the near side of the cage and then lands in the sector is a valid throw under TR 32.14 Note (iii). This game ends the trial instead, because the deflection cannot be modelled honestly. The rule itself is implemented and tested separately.
- The landing mark is the head's first contact point treated as a point, not a crater. Wind, latitude, altitude, temperature and the rotation of the Earth are not modelled. For scale, Horváth and colleagues compute that moving a men's world-record throw from the poles to the equator adds up to 45 cm, 2 m/s of wind up to 70 cm, and 3500 m of altitude up to 1 m, while the true Coriolis effect across the whole sector is 1.5 cm.
- The eleven rival throwers are invented. No real athlete is represented, and their marks are drawn from a seeded random distribution, not from any results database.
- The thrower's body is drawn as a jointed stick figure posed from the orbit. It is not a biomechanical model of a human, and nothing about the drawing feeds back into the physics.
Provenance
Every constant and every claim in this build carries a tag. DOCUMENTED was quoted from a source that was opened; DERIVED was computed from documented values by a stated rule; MEASURED is a number this engine produced and its harness re-measures on every run; CALIBRATED was fitted so the model reproduces a published observable; RECONSTRUCTED is my own choice, with nothing published to pin it.
Of 192 items: 133 documented (69.3%), 24 derived, 14 measured, 6 calibrated, 15 reconstructed (7.8%).
Those counts are recounted from js/provenance.js by
tools-harness-page.js and checked against this paragraph and against
CREDITS.txt, so they cannot drift. The full list is under
.
One caution about that tally, stated here rather than buried: three of the documented items — standard gravity, sea-level air density and viscosity, and the textbook smooth-sphere drag curve — are general reference values, not things taken from a document opened for this build. They are marked as such in the source list so the percentage is not flattered.
Credits
The hammer throw is a World Athletics field event with a history going back to the Tailteann Games; it has been on the Olympic programme for men since 1900 and for women since 2000. This page is an independent reimplementation of the sport, built from the published rules and the published biomechanics. It is not affiliated with, endorsed by or connected to World Athletics, and it reproduces no third-party code, artwork or data files.
The men's world record is 86.74 m, set by Yuriy Sedykh at the European Championships in Stuttgart on 30 August 1986. The women's is 82.98 m, set by Anita Włodarczyk on 28 August 2016.
Renderer, engine, geometry and art are hand-written for this build; no third-party library of any kind is loaded or bundled. Full source list and licence in CREDITS.txt and LICENSE.txt.