Gravitional Potential Energy and Mass of Earth

[Tirokai]

This question is 12.59 from University Physics 11e.

"There are two equations from which a change in the gravitational potential energy U of the system of mass m and the Earth can be calculated. One is U=mgy. The other is U=GMm/r (M=mass of earth). The first equation is correct only if the gravitational force is a constant over the change in height delta-y. THe second is always correct. Actually, the gravitational force is never exactly cosntant over any change in height, but if the variation is small, we can ignore it. Consider the difference in U between a mass at the Earth's surface and a distance h above it using both equations, and find the value of h for which mgy is in error by 1%. Express this value of h as a fraction of the Earth's radius, and also obtain a numerical value for it."

The correct answer, from the rear of the book, is 0.01Re and 64 km.

My strategy, thus far, was to use the two gravitational potential energy formulae in an error formula. So I tried several different variations on (mgh-(-GMm/(Re+h)^2+Gmm/(Re)^2))/mgh=.01

Re= radius of Earth M=mass of earth

Having set up the formula, I used the solve function of a TI-89 to punch them out, all of them eventually coming out to be some random number in the millions or tens of millions.

Any help is greatly appreciated :D

 

[Seiya]

In your formula you are squaring the R, why? For potential energy you don't square the r... :(

 

[Tirokai]

You know what, I accidentally copied that from a formula someone else gave me (a physics major, too! shame on them) and so let me update the formula I used.

(mgh-((-GMm/Re+h)+(GMm/Re)))/(mgh)=.01

I simplified that down to

((gRe^2+GMe)h+(gRe)h^2)/((gRe^2)h+(gRe)h^2)=.01

Which gave me some bogus answer like -1.3x10^7

Thanks for the response, at any rate.

 

[Seiya]

.01 = (mgh - accepted value)/accepted value

thats it... you have an idea how to find the accepted value i think so that shouldn't eba problem... but i still see those squares there... there shoudnt be squares anywhere... if I am wrong please prove me wrong because I've got a test on this and waves tomorrow (curses at waves) :P