There are two equations from which change in gravitaton potential energy of a system can be calculated.
ΔEg = mgh and the other Eg = GmM/R
The first equation is only correct if the gravitational force is constant over a change in height h. The second is always correct. Consider the change in Eg of the mass m when it is moved away from the earth's surface to a height h using both equations, and find the value of h for which the equation ΔEg=mgh is in error by 1%. Express this value of h as a fraction of the earths radius and obtain the numerical value.
m = mass of object M = mass of earth Re = 6.38x10^6meters
ΔEg1= -GmM/(Re+h) + GmM/Re
The Attempt at a Solution
Fg = GmM/(Re+h)^2
mg = GmM/(Re+h)^2
g = GM/(Re+h)^2
ΔEg2 = mGMH/(Re+h)^2
So i have to find the the relationship between these equations
so i say
I)ΔEg1/ΔEg2 = 100/101
II)ΔEg1/ΔEg2 = 100/99
where 100 represents the one which does not have the fraction of error percent and 101 and 99 represent the one with the fraction of error percent
When i solve I my answer is
h = -Re/101
which yields a negative number so this is not the answer.
II) When i solve for this
h = Re/99
h= 64444.44 m
When i double check these answers with my teacher she doesn't seem to have the same numbers so i'm just curious what i'm doing wrong?
I know there are 2 other cases i didn't check
III)ΔEg1/ΔEg2 = 99/100
IV)ΔEg1/ΔEg2 = 101/100
where III) yields h = Re/100
IV) h = -Re/100
h = -63800
But i know III) and IV) are not the case because that would mean that Eg = -GmM/Re is the equation off by a fraction of a percent.
Anyways any help is appreciated.