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## Homework Statement

Hi everybody! I'm stuck at a problem I was given in my special relativity theory class, can anybody help?

A golfer is on the North Pole, on a flat surface perpendicular to the Earth axis and without friction. He places himself at distance L from the pole and hits the ball exactly to the North Pole with an initial velocity v. Calculate at which distance from the hole the ball rolls, in which you must solve the Newton's equation of motion in the coordinate system rotating with the earth.

Tip: use Cartesian coordinates (x

_{1},x

_{2},x

_{3}) with x

_{3}along the Earth axis and write the equation of motion for the complex combination z = x

_{1}+ ix

_{2}.

## Homework Equations

Coriolis acceleration, coriolis force

## The Attempt at a Solution

So with a few students we managed to get an equation of motion, but then we're stuck because of the complex conjugate number:

[tex]

\ddot{r} = - \vec{ω} \times (\vec{ω} \times \vec{r}) - 2(\vec{ω} \times \dot{\vec{r}}) \\

\implies \ddot{r} + 2(\vec{ω} \times \dot{\vec{r}}) + \vec{ω} \times (\vec{ω} \times \vec{r}) = 0 \\

(\ddot{r_x}, \ddot{r_y}, 0) + 2 (- \dot{r_y} ω_z, \dot{r_x} ω_z, 0) - (r_x ω_z^2, r_y ω_z^2, 0) = 0 \\

\mbox{we combined the two equations (for x and y) and replaced the } r_x + r_y \mbox{ by } x_1 + ix_2 \\

\implies \ddot{z} + 2ω_z \dot{\bar{z}} - ω_z^2 z = 0

[/tex]

What do you guys think to begin with? The complex conjugate is strange, it's making the differential equation really hard, too hard for our level. We've checked everywhere but couldn't find a mistake. Any suggestions?

Thanks a lot in advance!

Julien.