The Verification Venue · pointed at a thing the textbook gets wrong twice
The Angle the Body Won't Let You Throw
Everyone learns the best launch angle is 45°. The physics-literate correct it: lower, because you release above the ground. Both are wrong for a human body. The real reason is that you launch faster when you launch flat — and the speed you lose by aiming higher costs more range than the angle ever buys back.
Nicholas Linthorne measured this. He filmed real shot putters and long jumpers throwing and jumping over a wide spread of angles, and found the same thing every time: the release speed an athlete can produce falls as the launch angle rises. A long, fast run-up hands you a flat, fast launch; turning that into a steep one is slow. Drag the angle below and watch it happen — the launch visibly slows, and the distance peaks well before it ever reaches 45.
Launch angle
34°
drag the slider
Launch speed at this angle
13.8 m/s
measured v(θ)
Distance (this angle)
— m
peak is elsewhere
As you steepen past the peak, the launch speed in the middle readout keeps dropping — that fall is why the distance curve crests so far left of 45.
Now switch the real effects on one at a time and watch the peak of the distance curve slide left:
The grey curve is the schoolbook: hold the speed fixed, release from the ground, and the best angle is exactly 45°. Switch on release height — the shot leaves a putter's hand about 2 m up — and the peak slides only a little, to about 42°. That is where most "well, actually" answers stop. Then switch on the measured speed–angle curve, and the peak collapses: to about 31° for the shot put, about 21° for the long jump — right where the athletes actually compete.
The check — every number recomputed in front of you
The optimum angles below are found by sweeping the launch angle and maximising the projectile range R = (v²·sin2θ)/(2g)·[1 + √(1 + 2gh/(v²·sin²θ))] live, with the launch speed read from the measured v(θ) fit at each angle. Switch events above and these update.
| model | what's added | optimum θ | peak dist. |
|---|
The launch speed the model reads at the optimum, and at 45°, for the current event:
Run the offline mirror: node research/not-forty-five/verify-not-forty-five.mjs — it re-derives all of these from the same equation and the papers' coefficients (14/14).
What's measured, what's chosen, and what this model can't see
What the papers measured. Linthorne (2001) filmed five college shot putters (Maheras' 1995 data) throwing over a wide angle range; for the example athlete here (his Athlete 1) the measured release was 13.8 m/s at 34.1° from 2.11 m, and his published optimum — the angle that maximises the throw given how his speed falls with angle — was 31.3°. Linthorne, Guzman & Bridgett (2005) filmed three long jumpers and found the optimum take-off angle is about 21°, matching elite jumpers. The dominant effect in both is the speed–angle trade-off, not release height.
The free choices we made. For the shot we use a linear v(θ) — Linthorne's own validated short-cut, which he reports lands within 0.3° of his full force model — and we set its slope (0.0876 m/s per degree) so the range optimum reproduces his published 31.3° for this athlete. For the long jump we use Linthorne 2005's empirical fit v(θ) = (10.5−3.4)/(1+(θ/35)⁴)+3.4 and the CM heights 1.29 m (take-off) / 0.65 m (landing). These are regressions for specific athletes, with scatter — not universal constants. Each athlete has their own curve and their own optimum.
The numbers vary by athlete. Across Linthorne's five shot putters the calculated optima ran 28–34°, and across the long-jump literature the optimum sits around 21° but individual take-off angles span roughly 15–27°. And note the honest gap: the calculated shot-put optimum (~31°) is a little below the observed average competition angle (~37°) — the curve is flat near its peak, so athletes lose almost nothing by throwing a few degrees high, and many do. The matching is athlete-by-athlete to each putter's own preferred angle, not to the crowd average.
What the model can't see. Air drag on the shot is real but small — Linthorne reports it costs about 8 cm of distance and shifts the optimum by ~0.1°, so we omit it (the long jump is treated as free flight of the centre of mass). The long-jump model is a point projectile: it follows the centre of mass and ignores in-flight technique (the hitch-kick), which sets landing posture but not the CM parabola, so the CM range here (~7.8 m) is a little under the official ~8 m a feet-forward landing adds. None of this changes the headline: the speed–angle trade-off, not geometry, is what drags the best angle far below 45.