Physics primer¶

This page walks through the four forces that govern a pitched baseball's trajectory in pitchphys, what each one does, and what changes when you toggle it off. The goal is intuition, not derivation — for full equations and references, see SPEC §4 in the source repo.

The four forces¶

A baseball in flight experiences four physical effects in pitchphys:

Gravity — always pulls the ball downward.
Drag — opposes velocity; slows the ball.
Magnus lift — perpendicular to both velocity and spin axis; the source of pitch movement.
Spin decay (optional) — angular velocity shrinks over flight time.

simulate(pitch, forces=["gravity", "drag", "magnus"]) is the default. Toggling any subset on/off is a first-class educational feature — try forces=["gravity"] to see what a spinless pitch would do.

1. Gravity¶

\[\mathbf{F}_g = -m g \hat{\mathbf{z}}\]

The simplest force. With only gravity, the ball follows a parabola: \(z(t) = z_0 + v_{z,0} t - \tfrac{1}{2} g t^2\). Default g = 9.80665 m/s².

What "rising fastball" really means. A fastball never literally rises in MLB conditions — gravity always pulls it down. But the Magnus force on a high-backspin pitch counteracts a portion of gravity's drop, so the ball ends up ~20 inches higher at the plate than a spinless equivalent. That delta — the induced vertical break (IVB) — is what makes the pitch look like it "rises" to the hitter.

pitchphys reports IVB as traj.induced_vertical_break_in. Positive = above the no-force projectile path at the same flight time.

2. Drag¶

\[\mathbf{F}_D = -\tfrac{1}{2} \rho \, C_D \, A \, |\mathbf{v}_{\text{rel}}| \, \mathbf{v}_{\text{rel}}\]

Drag points opposite the ball's velocity (relative to wind) and scales with \(|v|^2\). It slows the ball from release to plate, which both reduces the time gravity has to act and the time Magnus has to lift.

The drag crisis. \(C_D\) is not constant. At low spin, \(C_D\) drops sharply across \(\text{Re} \approx 75\text{k} \to 175\text{k}\) — a phenomenon called the drag crisis. The default LyuAeroModel captures this; the simpler ConstantAeroModel uses \(C_D = 0.35\) throughout.

Drag and air density. Drag scales linearly with air density \(\rho\). A baseball at Coors Field (1.00 kg/m³, ~5,200 ft elevation) experiences ~18% less drag than at sea level (1.225 kg/m³) — which is why fly balls carry farther in Denver. See Environment & weather.

3. Magnus lift¶

\[\mathbf{F}_M = \tfrac{1}{2} \rho \, C_L \, A \, |\mathbf{v}_{\text{rel}}|^2 \, \widehat{(\boldsymbol{\omega} \times \mathbf{v}_{\text{rel}})}\]

The Magnus force is perpendicular to both velocity and spin axis, pointing in the direction \(\boldsymbol{\omega} \times \mathbf{v}_{\text{rel}}\). The magnitude depends on the spin factor \(S = R \, |\boldsymbol{\omega}_\perp| \,/\, |\mathbf{v}|\).

Backspin (axis perpendicular to velocity, pointing "right" in the catcher view): Magnus points up. Fastballs.
Topspin: Magnus points down. Curveballs.
Sidespin: Magnus points left or right. Sliders, sweepers.
Gyrospin (axis aligned with velocity): Magnus is zero.

The lift coefficient \(C_L\) is what each aerodynamic model in pitchphys.aero specifies:

LyuAeroModel (default): empirical fit from Lyu 2022 wind-tunnel data.
NathanLiftModel: Sawicki–Hubbard–Stronge bilinear: \(C_L = 1.5 S\) for \(S < 0.1\), \(C_L = 0.09 + 0.6 S\) for \(S \ge 0.1\).
SimpleMagnusModel: pedagogical \(C_L = \min(a S, C_{L,\max})\).

See Aerodynamic models for tradeoffs.

4. Active spin vs gyro spin¶

Not all spin produces movement. Only the component of \(\boldsymbol{\omega}\) perpendicular to velocity contributes to the Magnus force. The parallel component (gyro spin) doesn't move the ball.

pitchphys exposes this as the active_spin_fraction parameter on PitchRelease.from_mph_rpm_axis(...):

`active_spin_fraction`	Meaning
`1.0`	All spin is perpendicular to velocity — full Magnus break.
`0.95`	95% active — typical of a high-spin 4-seam fastball.
`0.30`	30% active — gyro slider archetype.
`0.0`	Pure gyro — no Magnus break (in theory).

In practice, even a 0% active-spin pitch picks up a small amount of Magnus mid-flight, because gravity bends \(\mathbf{v}_{\text{rel}}\) while \(\boldsymbol{\omega}\) stays constant in the world frame, so the perpendicular component grows slightly over flight.

Spin rate is not movement. A 2,500 rpm gyro slider has the same total spin as a 2,500 rpm 4-seam fastball — and yet the slider has roughly no vertical break. This is why Statcast publishes "Active Spin %" as a first-class metric alongside spin rate. Try the Active Spin / Gyro Streamlit page to see it interactively.

5. Spin decay (optional)¶

\[\boldsymbol{\omega}(t) = \boldsymbol{\omega}_0 \, e^{-t / \tau}\]

Real baseballs lose ~10–20% of their spin over the 0.4 s flight to the plate. By default simulate(...) uses spin_decay_tau_s=1.5, giving ~23% decay over a typical fastball flight. Pass spin_decay_tau_s=None to disable (matches SPEC §8.1 strict reading and Nathan 2008's fit assumption — important when comparing to specific published numbers).

What changes when you toggle a force off¶

Try this in the Pitch Playground or as Python code:

from pitchphys import simulate
from pitchphys.presets import four_seam

pitch = four_seam(speed_mph=95, spin_rpm=2400)

# Each successive simulation adds one more force.
g_only       = simulate(pitch, forces=["gravity"])
g_drag       = simulate(pitch, forces=["gravity", "drag"])
g_drag_magnus = simulate(pitch, forces=["gravity", "drag", "magnus"])  # default

print(f"Gravity only:        IVB {g_only.induced_vertical_break_in:+.1f} in")
print(f"+drag:               IVB {g_drag.induced_vertical_break_in:+.1f} in")
print(f"+Magnus (full):      IVB {g_drag_magnus.induced_vertical_break_in:+.1f} in")

Each force adds an additional ~5–20 inch deviation from the previous trajectory. The Magnus contribution alone is what distinguishes a high-IVB fastball from a sinker thrown at the same speed and spin rate.