Golfing on North Pole (equation of motion)

[JulienB]

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 (x₁,x₂,x₃) with x₃ along the Earth axis and write the equation of motion for the complex combination z = x₁ + ix₂.

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:

$$\ddot{r} = - \vec{ω} \times (\vec{ω} \times \vec{r}) - 2(\vec{ω} \times \dot{\vec{r}}) \\ \implies \ddot{r} + 2(\vec{ω} \times \dot{\vec{r}}) + \vec{ω} \times (\vec{ω} \times \vec{r}) = 0 \\ (\ddot{r_x}, \ddot{r_y}, 0) + 2 (- \dot{r_y} ω_z, \dot{r_x} ω_z, 0) - (r_x ω_z^2, r_y ω_z^2, 0) = 0 \\ \mbox{we combined the two equations (for x and y) and replaced the } r_x + r_y \mbox{ by } x_1 + ix_2 \\ \implies \ddot{z} + 2ω_z \dot{\bar{z}} - ω_z^2 z = 0$$

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.

 

[mfb]

You are probably allowed to neglect higher-order effects, which means you know $\dot z$ (and also its complex conjugate) when calculating $\ddot z$.

Mathematically, the complex conjugate is just a minus sign for x2.

This problem is so much easier in non-rotating coordinates... at least you can use them to check the result.

 

[JulienB]

@mfb thanks for your answer, but I am afraid it doesn't help much. I know what the conjugate of a complex number is, but I can't solve the differential equation with it inside. I tried to make it disappear somehow but no luck so far! :)

 

[mfb]

I don't think you need a full analytic solution.

But thinking about it, are you sure it is right with the complex conjugate? There should be a factor of i somewhere.

 

[JulienB]

@mfb Well I need to find an equation of motion relative to the Earth axis I imagine, right? I forgot to give an indication: we can check the formula we get with L = 200m and v = 30m/s and we should get ca. 10 cms.Julien.

 

[JulienB]

Also, another student told me we can ignore the centrifugal acceleration (why??). That simplifies the equation a little bit, but still doesn't solve the complex conjugate issue.

 

[TSny]

Also, another student told me we can ignore the centrifugal acceleration (why??).

Consider the ratio of the magnitude of the centrifugal term to the magnitude of the Coriolis term for typical numbers.

That simplifies the equation a little bit, but still doesn't solve the complex conjugate issue.

Note mfb's comment in post 4. The complex conjugate term is not written correctly. There's been a mishandling of a factor of i.