How to Calculate Escape Speed Using Newton's Gravitational Law

[mitch_nufc]

An object is fired vertically upwards from the surface of a planetary body; it moves under the action of Newton’s Gravitational Law, without
resistance, so the equation is z'' = -gR^2 / (z + R)^2 . Find the relation between v = z' and z and use this model, and the relation that you have
found, to obtain a numerical estimate for the escape speed on the surface of the Earth.

What I've done so far is transformed z'' to vz' using the chain rule.
Next vz' = f(z)
=> 1/2*v^2 = int( f(z)dz + C )
now solve v(z) = dz/dt
=> int (dt) = int(1/v(z)dz)
So I got (gR^2)/(z+R) = (v^2)/2
(2gR^2)/(z+R) = v^2

This is where I got and I don't know where to go next, I don't even know if it's right? Could someone attempt this for me please? Thanks

 

[Cyosis]

You arrived at the right expression, however you ignored the integration constant. How is z defined, is it the distance from the surface to the Earth or from the center of the earth? Looking at your formula I would guess from the surface of the earth. This would give you a boundary condition v(0)=0.