How high would the end of a very long and level jetty be above sea level?

Question

If you built a perfectly level jetty, starting at the shore and going out to sea, presumably its end would get higher and higher above sea level the longer the jetty was. Assuming the Earth is uniformaly spherical, (which it is not), how high above sea level would the end of the jetty be if the jetty was, say, 50 miles long? Or, how long would the jetty have to be for the end to be 1,000 feet above sea level?

If you built a perfectly level jetty, starting at the shore and going out to sea, presumably its end would get higher and higher above sea level the longer the jetty was. Assuming the Earth is uniformaly spherical, (which it is not), how high above sea level would the end of the jetty be if the jetty was, say, 50 miles long? Or, how long would the jetty have to be for the end to be 1,000 feet above sea level?

I am assuming the radius of the Earth is 4,000 miles. This is not 100% accurate but close enough for my question!

Wow - thanks for the answers so far. Yes, my wording was sloppy. I am not a mathematician, so wasn't sure what word I should have used. Now I know, STRAIGHT is what I meant. Anyway, thanks so far - very impressed.

Answer 1

So, lets make the radius of the Earth (4,000 miles as you stated) r. Then 50 miles would be r/80.

Now consider a circle at the origin of radius r. Since we are assuming the Earth is perfectly spherical (I am assuming that the jetty is built along a great circle) we can put the start of the jetty at the top of the circle (0,r). Now consider the point that is r/80 to the right of y axis, but is still on the circle, this will be represented by (r/80,y) solving for y:

(r/80)^2+y^2=r^2 ==>> y^2=r^2(1-1/80^2)=r^2(6399/6400).
Thus y=±(r/80)*(√6399), we only want the positive y value in this case, so y=(r/80)√6399 ≈ 3999.687488. Therefore the height of the jetty above this point would be (approx.) 4000-3999.687488 =0.312512 miles, or 1650.063360 feet.

If you want the end to be 1000 feet (or 25/132 miles) above the ocean, you would want to solve for x in (x,4000-25/132).
This would give you x≈38.92448644 so the jetty would need to be about 38.92448644 miles long.

But you should be careful of your wording. Level means tangent to the Earth, thus the jetty would wrap around the Earth and be the same distance above the surface it started, if it were perfectly level. What I solved for is if the jetty was perfectly straight (and extended along a great circle, if this isn't true then the answer would change depending on which direction you extend the jetty).

Answer 2

Under the word "straight" I understand "parralel to the surface of earth"
In that case any length will do, if your jetty is a 1 000 feet above sea level at the start

Answer 3

.Assume then the earths radius, the length of your jetty, Get a huge piece of paper and a scale ruler. Draw a circle to desired scale. Choose point on circle edge and draw a line perpendicular to the radius line to the predetermined length of your jetty and draw a perpendicular line from its edge to the curvature of the earth below. Where they intersect, connect with origin point and use Pythagoras.

I'm sure there are easier ways to calculate this, but I'm no mathematician.

Answer 4

not an answer.. but a comment... your jetty is NOT level.. it is STRAIGHT... level refers to it being perpendicular to a line drawn to the center of the earth... so it would STAY at the same height above the water... and be circular.. to fit the curve of the earth.

Answer 5

What are you taking as the dimensions of planet earth!