Golfing on North Pole (equation of motion)

In summary, the conversation discusses a problem in special relativity theory involving a golfer hitting a ball from the North Pole and calculating its distance from the hole on a rotating Earth. The conversation also addresses the use of Cartesian coordinates and the complex conjugate number in solving the equation of motion. Suggestions are given for simplifying the equation and checking the result with specific values.
  • #1
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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 (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:

[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.
 
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  • #2
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.
 
  • #3
@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! :)
 
  • #4
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.
 
  • #5
@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.
 
  • #6
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.
 
  • #7
JulienB said:
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.
 
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1. How would the equation of motion for golfing on the North Pole be different?

The equation of motion for golfing on the North Pole would be different because the North Pole is located at the Earth's axis, which means it experiences a different rotational velocity and gravitational pull compared to other locations on Earth. This would affect the trajectory and distance of a golf ball, as well as the force needed to hit it.

2. Would the Coriolis effect have an impact on golfing at the North Pole?

Yes, the Coriolis effect would have a significant impact on golfing at the North Pole. This phenomenon, caused by the Earth's rotation, would cause a golf ball to veer to the right in the Northern Hemisphere and to the left in the Southern Hemisphere, making it more challenging to hit a straight shot at the North Pole.

3. How would the colder temperatures at the North Pole affect golfing?

The colder temperatures at the North Pole would affect golfing in a few ways. Firstly, the colder air is denser, which means it would provide more resistance to the golf ball, causing it to travel a shorter distance. Additionally, the colder temperatures could also affect the elasticity of the golf ball and the clubs, making them less bouncy and reducing their performance.

4. How would the lack of daylight at the North Pole affect golfing?

The lack of daylight at the North Pole would make it challenging to play golf at certain times of the year. Golfers would have to rely on artificial lighting, which could affect their depth perception and ability to accurately judge distances. The low visibility could also make it challenging to find and retrieve golf balls.

5. Is it possible to play golf at the North Pole?

Yes, it is possible to play golf at the North Pole, although it would be a unique and challenging experience. Golfing equipment and clothing would need to be specially designed to withstand the extreme temperatures and conditions. Additionally, the course would have to be constructed on a frozen surface, and players would have to adapt to the unique environmental factors, such as the Coriolis effect and lack of daylight.

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