What's the minimum. length of the arc on the cicumference of the earth,where a curvature of the horizon is appearent?
OR---What's the max.length of cord,drawn
on the cicumference of the earth,where
length of the cord appears same as arc?
Follow up 2nd question---At what minimum
height above earth the curvature is noticable?
To find the minimum length of the arc on the circumference of the Earth where a curvature of the horizon is apparent, we can use the formula for the curvature of a sphere. The formula is:
C = 2πr(theta/360)
Where C represents the length of the arc, r is the radius of the Earth, and theta is the angle in degrees that represents the extent of the arc.
The radius of the Earth is approximately 6,371 kilometers (or 3,959 miles). We want to find the minimum length of the arc where the curvature can be noticeable, so we can assume a reasonable distance for the angle theta to be, for example, 1 degree or 0.5 degrees.
Using these values, we can calculate the length of the arc using the formula mentioned above.
For example, with theta = 1 degree:
C = 2 * 3.14 * 6371 * (1/360)
C ≈ 111.32 kilometers (or 69.2 miles)
So, the minimum length of the arc on the circumference of the Earth, where a curvature of the horizon is apparent, is approximately 111.32 kilometers (or 69.2 miles).
For the second question, the maximum length of a cord drawn on the circumference of the Earth where the length of the cord appears the same as the arc can be found using a similar approach.
Let L be the length of the cord and S be the length of the arc on the circumference of the Earth.
Since the length of the cord appears the same as the arc, we have:
L = S
We can rearrange the formula for the circumference of a circle to solve for the length of the arc:
S = 2πr(theta/360)
Substituting L for S, we get:
L = 2πr(theta/360)
To find the maximum length of the cord, we can choose a value for theta that is close to 360 degrees (a full circle).
For simplicity, let's assume theta = 359 degrees.
L = 2π * 6371 * (359/360)
L ≈ 39,999 kilometers (or 24,854 miles)
Therefore, the maximum length of a cord drawn on the circumference of the Earth, where the length of the cord appears the same as the arc, is approximately 39,999 kilometers (or 24,854 miles).
For the follow-up question on the minimum height above the Earth where the curvature is noticeable, it depends on various factors such as atmospheric conditions, personal visual acuity, and the observer's field of view. However, a generally accepted estimate is that the curvature is visible at an altitude of around 35,000 feet (or 10,668 meters) above sea level.