As a baseball (or any sphere or cylinder) travels, it drags a thin layer of air with it, causing a difference in pressure on each side of the ball. The ball moves toward the area of lower pressure. The faster the spin, the greater the pressure differential, and the greater the movement. A curveball breaks downward because its topspin creates a downward Magnus effect. A fastball with a high spin rate appears to rise—even though it really just falls less than a lower-spin pitch—because its backspin produces a Magnus effect that pushes the ball up, opposing gravity.
