What is the Magnus Coefficient for a Golf Ball? When researching the magnus force I have often come across this formula to calculate it:
$$ F=\vec S\cdot (\vec{\omega} \times \vec{v})$$
However I have been unable to find any source giving the value of $S$. Different sources also have different names for it such Magnus coefficient or air resistance coefficient across the objects surface (then again those might be the same thing). Does anybody know where I can find the value for $S$ for a golf ball?
 A: 
to calculate the factor $S$ You can use those equations:
$$\vec{v}_A=\vec{v}_0+\vec{\omega}\times \vec{r}\tag 1$$
$$\vec{v}_B=\vec{v}_0-\vec{\omega}\times \vec{r}\tag 2$$
and the bernoulli equation :
$$p_A+\frac{\rho}{2}\vec{v}_A^2=p_B+\frac{\rho}{2}\vec{v}_B^2$$
or
$$p_B-p_A=\Delta p=\frac{\rho}{2}\left(\vec{v}_A^2-\vec{v}_B^2\right)\tag 3$$
with equations (1) ,(2) and (3) we get
$$\Delta p=2\rho\vec{r}\cdot \left(\vec{v}_0\times \vec{\omega}\right)$$
where $\rho$ is the air density.
the force is then $F=\int \Delta p\,dA =\int 2\rho\,\vec{r}^T\,dA\,\left(\vec{v}_0\times \vec{\omega}\right)=(\vec{S})^T\,\left(\vec{v}_0\times \vec{\omega}\right)$
with: $(\vec{S})^T=\int 2\rho\,\vec{r}^T\,dA$
where $dA$ is the sphere area element
$dA=r^2\sin(\theta)d\theta\,d\phi$
with the sphere position vector :
$$\vec{r}=a\,\left[ \begin {array}{c} \cos \left( \theta \right) \sin \left( \phi
 \right) \\ \sin \left( \theta \right) \sin \left( 
\phi \right) \\ \cos \left( \phi \right) 
\end {array} \right] 
$$
where $a$ is the sphere radius , $\theta$ the azimuth angle and $\phi$ the polar angle.
we assume that the wind blows toward the $y$ direction:
$$\vec{v}=\left[ \begin {array}{c} 0\\ v_{{0}}
\\ 0\end {array} \right] 
$$
and the rotation of the ball is around the $z$ axis .
$$\vec{\omega}= \left[ \begin {array}{c} 0\\ 0\\ 
\omega_z\end {array} \right] 
$$
so we get for $S$:
$$\vec{S}(\theta\,,\phi)=\rho\,a^3\, \left[ \begin {array}{c} 2\,\sin \left( \theta \right)  \left( -\frac 12\,
\cos \left( \phi \right) \sin \left( \phi \right) +\frac 12\,\phi \right) 
\\-2\,\cos \left( \theta \right)  \left( -\frac 12\,\cos
 \left( \phi \right) \sin \left( \phi \right) +\frac 12\,\phi \right) 
\\ -\theta\, \left( \cos \left( \phi \right) 
 \right) ^{2}\end {array} \right] $$

