Solving a Problem with Non-Constant g: Expanding a Maclaurin Series

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Homework Help Overview

The problem involves an object dropped from a height h above Earth, where the acceleration due to gravity, g, is not constant. The original poster attempts to derive the speed at which the object hits the ground, considering the effects of varying gravity and suggesting the use of a Maclaurin series expansion.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of conservation of energy and the implications of non-constant g. The original poster questions the applicability of mgh due to the variable nature of g. Others suggest alternative expressions for g, such as mMG/(r+h), and inquire about the calculations leading to discrepancies in results.

Discussion Status

The discussion is ongoing, with participants sharing attempts and results. Some guidance has been offered regarding the expression for g, but there is no explicit consensus on the correct approach or final answer yet.

Contextual Notes

Participants note the challenge of deriving the correct expression due to the non-constant nature of g and the specific requirement to expand a Maclaurin series. There is an emphasis on the need for detailed calculations to identify errors in reasoning.

didinyc
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Homework Statement


Suppose an object is dropped from height h above Earth where h<<R, but is large enough so g, the acceleration due to gravity, is NOT constant! Show that speed with which it hits the ground, neglecting friction, is approximately, v= sqrt2gh *(1-(h/2R))

Hint: you will need to expand an expression in Maclaurin series.

The Attempt at a Solution


I tried to use conservation of energy and find g by using the gauss' law i got a close result but i didnt get the 2 in the last solution! Then I realize I cannot use conservation of energy because the g is not constant so i cannot use mgh! any ideas how could i solve this proble
 
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welcome to pf!

hi didinyc! welcome to pf! :smile:
didinyc said:
… because the g is not constant so i cannot use mgh!

so use mMG/(r+h) :wink:
 


tiny-tim said:
hi didinyc! welcome to pf! :smile:so use mMG/(r+h) :wink:

thank you! I tried what you said too but the answer that i got is wrong.It is really close answer except the 2 in front of the R in the v equation! so this is what i got as my result: v= sqrt2gh *(1-(h/R))!
 
hi didinyc! :smile:

(just got up :zzz: …)

(have a square-root: √ :wink:)

show us your full calculations, and then we'll see what went wrong! :smile:
 

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