Newton's second law applied perpendicular to the ball's flight path. $F$ is lateral force from asymmetric airflow, $m$ is baseball mass ($~0.145$ kg or $10^{-2}$ slugs), $x$ is lateral displacement. Further Reading: - [The Physics of Baseball - Alan M. Nathan - University of Illinois](https://baseball.physics.illinois.edu/knuckleball.html) - [KNUCKLEBALLS - Rod Cross - May 2013](https://www.physics.usyd.edu.au/~cross/KNUCKLEBALLS.htm) At most angles, there is a steady lateral force (up to $\sim 0.1$ lbs) determined by the stitch position. At specific "critical" angles (approx. $\theta = 52^{\circ}$ and $310^{\circ}$), the lateral force oscillates spontaneously. This occurs when a stitch is located precisely at the boundary layer separation point, causing the wake to flip-flop between the front and rear of the stitch The authors find that: - Slow spin beats zero spin. - Zero spin $\Rightarrow$ constant force $\Rightarrow$ predictable curve. - Slow spin $\Rightarrow$ seams rotate through critical zone $\Rightarrow$ erratic force. - Too much spin $\Rightarrow$ ball inertia prevents meaningful deflection. The authors model the lateral motion using the steady-state solution for a periodic force $F(t) = F_0 \sin(\omega t)$. The resulting displacement equation is: $$x = \frac{F_0}{m\omega^2} \sin(\omega t + \pi)$$ The magnitude of lateral displacement is inversely proportional to the square of the frequency ($1/\omega^2$). Only forces from a slow rotation last long enough to push the ball significantly off its path. The total distance a knuckleball curves is independent of how fast it is thrown. A faster pitch has a larger deflecting force, but less time for that force to act, and these two factors cancel each other out perfectly.  ***knuckleball:*** a slow pitch that has virtually no spin and moves erratically, typically made by releasing the ball from between the thumb and the knuckles of the first joints of the index and middle finger. ## TL;DR This 1975 paper by Watts and Sawyer investigates the aerodynamic mechanism behind the erratic motion of the baseball pitch known as the "knuckleball." The authors put baseball in a wind tunnel to explain knuckleball aerodynamics. They identified two lateral force mechanisms: 1. **Steady seam asymmetry:** No matter how the ball is oriented, the seams usually create a small, steady sideways force. 2. **The "critical seam" effect:** When a seam rotates into the separation zone (roughly 105–115 degrees from the front), the airflow can't decide which side to detach from. The wake jumps back and forth, flipping the lateral force about once per second. The authors found that total sideways deflection doesn't depend on how fast it's thrown - higher speed increases the aerodynamic force, but shortens the flight time in perfect balance. The secret is slow spin: about half a rotation is enough for the seams to drift into the separation zone and destabilize the airflow mid-flight. **The "break" is an illusion.** Constant lateral force produces a parabolic trajectory, so displacement grows as $t^2$. Half of all sideways movement happens in the final 30% of the flight. The batter's brain interprets this as a sudden break but the ball is simply following a parabola whose sideways motion accelerates with time.