Give a qualitative explanation, based on Bernoulli's equation, of how it is possible to throw a baseball in a "curve."

Bernoulli's equation is a fundamental principle in fluid dynamics that relates the pressure, kinetic energy, and potential energy of a fluid. The formula is given by: \(\text{constant} = P + \dfrac{1}{2} \rho v^2 + \rho gh\) where: - P is the pressure acting on the fluid, - \(\rho\) is the fluid's density (air, in this case), - v is the fluid's velocity, - g is the acceleration due to gravity, and - h is the height of the fluid.

A curveball is thrown in such a way that the baseball has a significant amount of topspin or backspin (depending on the desired curve direction). This spinning motion creates a pressure difference on the ball's surface as it moves through the air, causing the ball to curve.

The Magnus effect is the phenomenon responsible for creating the pressure difference on the spinning baseball's surface. When the baseball spins, the air on one side of the ball moves faster, while the air on the opposite side moves slower. According to Bernoulli's equation, the faster-moving air has a lower pressure, while the slower-moving air has a higher pressure. This pressure difference results in a net force on the ball, causing it to curve.

When a pitcher throws a curveball, they typically grip the ball with their index and middle fingers on the seam and apply force on the ball during the pitch to generate the desired spin. As the ball moves through the air, the Magnus effect comes into play, creating a pressure difference on the ball's surface. The side of the ball moving faster through the air experiences a lower pressure, while the opposite side experiences a higher pressure. As a result, the ball experiences a net force acting towards the low-pressure region, causing the ball to curve in the desired direction. This curved trajectory makes it challenging for the batter to anticipate the ball's position, making curveballs an effective pitching technique.