Projectile and gravitation problem

[fluidistic]

Homework Statement

A body is fired vertically from the ground of the Earth with a velocity $$v_0=1km/s$$. If the gravitational force is not considered as constant and supposing that the Earth is a sphere of radius $$6371km$$,
a)Find the maximum height reached by the body.
b)Compare the previous result with the same experience if g is a constant. (That is $$9.8 m/s^2$$.)

Homework Equations

$$r_{max}=\frac{1}{\frac{1}{r_0}-\frac{v_0^2}{2GM_E}}$$

The Attempt at a Solution

I found the formula above in my class notes. If I understand it well, it gives "$$r_{max}$$", which I interpret as the distance of a body from the center of the Earth. $$r_0$$ is the radius of the Earth and $$M_E$$ is the mass of the Earth.
I don't know how to find this formula so I think I will have to learn it by heart (sadly...). If you know how to reach to it, please let me know.
Now using the formula, I found that the projectile will reach $$6690.36km$$! Oh wait... this is the height from the center of the Earth, not the ground as I thought... So it's probably right then.
Because for the b) I get that it reach only $$51.020km$$, but it is from the ground of the Earth. Now that I think it makes about $$319 km$$ (if g is not a constant), I think it's too much, isn't it?

 

[Dick]

You are right, forget the formula. It's too specific to a particular problem. The concept you want is that the sum of the gravitational potential energy (PE) and kinetic energy (KE) is a constant. If you consider the gravitation force to be constant PE=mgr. If you use Newton PE=-G*M*m/r. 319km does seem like a bit much. I agree with your 51.02km for the constant g and g doesn't vary so much near the Earth.

 

[fluidistic]

Thank you very much. I'll try it tomorrow, if I have any problem I'll ask for further help, but I think all will be all right.