Unlike a rising fastball, which simply appears to rise, a curveball really does curve. However, by using first principles of Classical Mechanics, one cannot show why a curveball does what it does.
As a German named Gustav Magnus figured out in 1852, the reason curveballs curve is becuase the air flows differently on the side of the ball spinning toward the plate than the side spinning away from the plate.
Here's how it works:
The speed of the air at the top of the ball is equal to the lateral speed of the ball (from the pitch being thrown) minus the speed caused by the ball spinning (since the ball is spinning opposite to the way the air is flowing). The speed of the air at the bottom of the ball is equal to the sum of the two speeds.
Using Bernoulli's equation, which (among other things) shows that as velocity increases pressure decreases, one can see that the pressure at the top of the ball is therefore greater than the pressure at the bottom of the ball. So, the ball drops.
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