Exercise Ribbon
How do I calculate where a weight lands from space.?
If a weight is dropped from 4000 or 8000 or 16000 or 20000 miles above the equator from the ribbon of a space elevator which is in geostationary orbit I assume it has two velocities. One angular the other accelerating to earth with an increasing gravitational pull. What method do I use to calculate where it lands. I want to do this from varying heights up to near geosynchronous orbit so that I can plot a graph. Height dropped against distance landed from mooring point. Do I have to make multiple calculations averaging the gravitational pull for say each 4000 miles or is there a formula? For this exercise I am assuming the weight is not attached and not subject to air friction.
Not a trick question. I thought that if the radius was doubled the speed was increased proportionally. A weight on a string one meter long spinning at one revolution per minute has a lower speed that one on a two meter string spinning at one rev. per min.
Remember that conservation of rotational impulse applies in the rotating frame of Earths surface. So while you start your drop with zero velocity relative to Earths surface, during the drop in the rotation frame, you will gain horizontal velocity while your radius reduces – you will land ahead of the elevator, and from some distance away, you won’t even reach ground at all.
Essentially, you are working with orbits, and the text book functions about orbits is a good way to solve this problem. You know apoapsis radius and apoapsis velocity, you also know true anomaly (since it starts at apoapsis). Now you need to solve this for the true anomaly when you intersect Earth’s surface. (Solve ellipse function for radius = 6378 km)
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