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

[didinyc]

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

 

[tiny-tim]

welcome to pf!

hi didinyc! welcome to pf!

… because the g is not constant so i cannot use mgh!

so use mMG/(r+h)

 

[didinyc]

hi didinyc! welcome to pf! so use mMG/(r+h)

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))!

 

[tiny-tim]

hi didinyc!

(just got up :zzz: …)

(have a square-root: √ )

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