The Motion Of Objects Thrown Away From Earth

[Correia]

How can the function for distance over time be found for an object which is thrown away from Earth? Acceleration as a function of distance is

g = GM/r^2,

where GM is constant, so let c = GM, then

g = c/r^2.

Then I wondered, maybe the formula for distance can be found by solving the differential equation

r'' = c/r^2?

I have never studied differential equations, so I have no idea about it, even to whether I can really say r'' = c/r^2.

If someone can elucidate my mind, I shall be grateful.

 

[Simon Bridge]

Welcome to PF;

Have a look at:
http://home.comcast.net/~szemengtan/ClassicalMechanics/SingleParticle.pdf
The general approach is given in section 1.7: "Central Force Problem"

The basic technique should be familiar, you choose a coordinate system that makes sense (spherical polar) and construct a free body diagram for your mass... put $\vec{F}=m\vec{a}$ and solve the differential equation for the appropriate initial condition.

Were you thinking of the Earth specifically or a non-rotating spherical mass M radius R?
Did you intend to throw the mass m directly upwards? (i.e. radially outwards) or solve the ballistics problem for the situation where the projectile does not remain close to the surface?
(The rotation of the Earth, etc, affects the result.)

Note: $\vec{F}=-mg(r)\vec{r}/r: r>R$