# Distance a curveball travels?

I've seen some discussions regarding the movement of a spinning object, say a curveball. However, all have been largely qualitative. I was wondering if anyone has seen or worked through a calculation of how far a curveball moves laterally on its way from the mound to homeplate - even in an order-of-magnitude sense.

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The two physical effects that come into play here are the Magnus effect and the drag force:

$$\mathbf{F}_M= S \left( \mathbf{\omega} \times \mathbf{v} \right)$$ $$\left| \mathbf{F}_D \right| =\frac{1}{2} \rho \left| \mathbf{v} \right|^2 C_D A$$

the direction of the drag force is oriented opposite to the direction of motion. $\rho$ is the density of the fluid (air in the case of a baseball), v is the velocity of the curveball relative to the fluid, A is the cross section of the baseball.

$S$ and $C_D$ are two coefficient to be determined. Typically $S$ is in the range 0.1-0.5 and $C_D$ is 0.1 for a smooth sphere, and will be a little more for a baseball which is not perfectly smooth.

Knowing the initial speed of the ball you can do the math by yourself, or refer to many articles and book written about the topic. Two references you may be interested in are:

1) "The physics of baseball" by Robert K. Adair 2) "The effect of spin on the flight of a baseball" by Alan M. Nathan, available here: http://webusers.npl.illinois.edu/~a-nathan/pob/AJPFeb08.pdf

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