Determination of Earths radius with known latitude and sun travel times

In summary, at latitude 40, the Sun lies due south 140 seconds after your friend observes it on her meridian. The radius of the Earth is equal to the radius of the disc at latitude 40 multiplied by the cosine of 40 degrees.
  • #1
frozenguy
192
0

Homework Statement


You and a friend decide to determine the radius of the Earth. You synchronize watches; then your friend drives 50km due west, at latitude 40°. Each of you determines the time when the Sun lies due south -- on the meridian Your friend observes the Sun to be on her meridian 140 seconds after you observe the Sun on your meridian. What is your determination for the Earth's radius?


Homework Equations


rθ=s
θ=2π(140s/86400s)

The Attempt at a Solution


With those equations I can find the radius of the latitude 40° disc section.

If I can find the height h from the center of the Earth to the center of my latitude 40 disc, I can use Pythagorean theorem to find radius of the Earth R. Is this the correct way to go about this? If so, how do I find the straight line height from center of disc to center of earth?
 
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  • #2
Think, what latitude means? Draw a picture and show the radius of the disc at latitude 40.
ehild
 
  • #3
ehild said:
Think, what latitude means? Draw a picture and show the radius of the disc at latitude 40.
ehild
Hi thank you for responding.
I have done that. What do you want me to see? It is not the radius of the earth. It is just the radius of a flat disk.
 
Last edited:
  • #4
Firstly find the radius of the Earth of your latitude by considering the quantities you have. Personally I would approach this by using ratio's you know the length of a day, the time between you and your friend and finally the distance. Once you have this radius you need to assume the Earth is spherical and translate the radius to the equator.
 
  • #5
Thanks for responding.
I have the radius of the disc at lat 40.
How do I translate the radius to the equator?
 
  • #6
frozenguy said:
I have done that. What do you want me to see? It is not the radius of the earth. It is just the radius of a flat disk.

I do not want anything from you, it is you wanting to see the solution. Make a drawing of the Earth like the one I attach. Show both the radius of Earth (R) and the radius of disk (r).
Do you know the angles of the yellow triangle? How are they related to the latitude?


ehild
 

Attachments

  • Earth.JPG
    Earth.JPG
    8.3 KB · Views: 550
  • #7
ehild said:
I do not want anything from you, it is you wanting to see the solution. Make a drawing of the Earth like the one I attach. Show both the radius of Earth (R) and the radius of disk (r).
Do you know the angles of the yellow triangle? How are they related to the latitude?


ehild

I apologize... I didn't really mean what do you want in that way. I shouldn't of said it like that.

I had drawn it many times but never like that. I think I need some sleep.
So R=r/cos40. :)

Thank you for your help. And again I didn't mean to sound rude or non appreciative.
 
  • #8
It is all right. You are welcome.

ehild
 

1. How is the Earth's radius determined using known latitude and sun travel times?

The Earth's radius can be determined by using the formula: R = D√(1+tan²φ), where R is the Earth's radius, D is the distance between two points on the Earth's surface, and φ is the latitude of those points. This formula is derived from the fact that the Earth is not a perfect sphere, but rather an oblate spheroid.

2. What is the significance of using latitude and sun travel times in determining the Earth's radius?

Latitude and sun travel times are important in determining the Earth's radius because they provide the necessary data points for the formula to be applied. Latitude gives the angle of the Earth's surface at a specific point, while sun travel times help calculate the distance between two points on the Earth's surface.

3. Is this method of determining the Earth's radius accurate?

Yes, this method of determining the Earth's radius is accurate, as long as the data used is precise and the formula is applied correctly. However, it should be noted that the Earth is not a perfect sphere, so there will be some margin of error in the calculated radius.

4. Are there any limitations to this method of determining the Earth's radius?

One limitation of this method is that it assumes the Earth is a perfect oblate spheroid, which is not entirely accurate. Additionally, this method can only be used for points on the Earth's surface that have known latitudes and sun travel times, so it may not be applicable in all locations.

5. Can this method be used to determine the radius of other planets?

Yes, this method can be used to determine the radius of other planets, as long as the necessary data (latitude and sun travel times) is available. However, the formula may need to be adjusted based on the shape of the planet being measured.

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