Physical · everyday science · shown not told
Why Does a Curveball Curve?
Because a spinning ball drags a thin skin of air around with it, and throws that air sideways — so the air throws the ball the other way. It is one force, the same for every pitch, pointing wherever the spin says. Throw one and watch.
The knuckleball flutters because it doesn't spin. Everything else on a pitcher's card — the ride of a four-seam fastball, the tumble of a curve, the late sweep of a slider — is one effect, the Magnus force, aimed in different directions by the axis of the spin. Below, every trajectory is integrated live from the real aerodynamics: gravity, air drag, and the Magnus force, with the lift and drag coefficients Alan Nathan measured from tracked major-league pitches. Nothing here is drawn by hand.
The mound · throw a pitch
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Vertical break is the honest way to say "rise". It measures how far the pitch misses the spinless path upward — how much less it drops than a ball thrown at the same speed with no spin. A good four-seamer beats gravity by a foot and a half of drop. But look at Total drop: it is still positive. The ball always falls. It just falls less than the batter's eye, trained on a lifetime of spinless arcs, predicts — and the miss reads as rise.
Can a fastball actually rise?
Only if the upward Magnus force beats gravity outright — if Magnus vs weight climbs past 100% and stays there. Set the direction to pure backspin and drag the spin slider up. At a real elite fastball (2,900 rpm) the Magnus force is about four-fifths of the ball's weight; the pitch still drops. Lift first equals weight around 4,600 rpm — already double a normal fastball — but even there the ball keeps sinking, because drag bleeds off the very speed the lift feeds on. Not until roughly 5,900 rpm at 95 mph does the ball truly climb above the hand. Real arms top out near 3,000. So the rising fastball is real as an illusion and impossible as a fact.
There is one honest loophole, and the instrument shows it: the faster the pitch, the less spin true rise would take (about 4,600 rpm at 100 mph, 4,000 at 105). The catch is that no arm throwing 105 also spins the ball at 4,000 — the two never arrive together. Drop the velocity and push the spin, and you can make the ball rise on screen; you have simply left the envelope any human has ever thrown inside.
The check — every number above is computed, not asserted
Each pitch is integrated with a 4th-order Runge–Kutta step (Δt = 0.2 ms) under three forces, in exactly the form the physics of baseball uses:
with the coefficients measured, not invented:
- C_L = 1.120·S / (0.583 + 2.333·S), where S = Rω/v is the spin factor — Nathan's trajectory-calculator fit. It gives C_L = 0.22 for a fastball, 0.25 for a curve; an independent fit (Sawicki–Hubbard–Stronge, 0.09+0.6S) agrees to ~0.01.
- C_D = 0.297 + 0.0292·(rpm/1000) — Nathan's spin-dependent drag. The real per-pitch drag scatters from ~0.28 to ~0.58 with weather and seam orientation (a genuine drag crisis); this is the central fit, not a constant of nature.
- Regulation ball: 145 g, 23.2 cm circumference (MLB rule midpoints); flight ≈ 55 ft, release to plate; sea-level air 1.225 kg/m³, Coors ≈ 1.05.
The identical model, run offline, reproduces the paper's stated coefficients and the Statcast benchmarks — 16 of 16 checks pass in research/curveball/verify.mjs. Run it yourself.
What's idealised, said plainly: the instrument assumes 100% of the spin is useful — pure back/top/sidespin. Real fastballs waste some spin as a gyro (bullet-like) component, so a perfect model sits at the elite end of Statcast (~20 in of vertical break) rather than the league average (~16 in). Nathan's coefficient fits were tuned partly on batted balls over an overlapping speed range. And the exact spin for "true rise" is model-dependent — Nathan's fit says ~4,600 rpm at 95 mph, Adair's older model ~3,600; either way, no arm reaches it.