Calulate the distance of objects in Earth's orbit

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To calculate the distance of objects in Earth's orbit from observations on the surface, the orbital speed equation (v = SQRT(G*M/R)) and the acceleration equation (a = G*M/R^2) are essential. A trigonometric approach may be useful for determining distances, particularly through orbit determination methods based on ground station observations. The complexity of these methods varies depending on the available data, such as angles and radial velocity. Further research into specific observational techniques could enhance accuracy. Understanding these calculations is crucial for analyzing the velocity and acceleration of orbiting objects.
VinnyO

Homework Statement


I am working on a report dealing with the velocity and acceleration of objects in Earth's surface based on distance from the Earth and thus far I have used the orbital speed equation and the acceleration equation. To get dive deeper into the math I would like to attempt to calculate the distance of these objects based on observations from Earth's surface. I have yet to fine something on the internet.

Homework Equations


v = SQRT(G*M/R)
a = G*M/R^2

The Attempt at a Solution


I have figured that there is likely a trigonometric solution.
 
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Unless there's a particular scenario that you have in mind, it sounds like might want to investigate orbit determination methods. There are several that are based on observations from ground stations. The complexity depends upon what observables are available (just angles, angles and distance, radial velocity, etc.).
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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