Object falling with non-constant gravity

[lastoneinspac]

The exact problem reads: The acceleration of gravity g is a constant only for a limited
range of height differences. A better approximation, one that
might hold over a larger range of height differences, is that g decreases
linearly with height, g = go - hg', where h is the
height measured from the ground surface and s' is a (small) constant
of the appropriate dimensions. (a) Find the speed of a
dropped object as a function of height assuming it was dropped
starting from rest from a height ho. (b) Find the speed of
a dropped object as a function of time assuming it was dropped
starting from rest from a height ho.

What I've tried to do so far for part a was to integrate acceleration in terms of h. I'm not sure if that is even allowed, but I ended up getting v = g0h - 0.5g'h^2

Then on part b, I more or less got stuck trying to get a formula for h in terms of t, and I'm not quite sure where to start.

I may just be thinking about this incorrectly, so any input on the best method for starting this problem would be welcome.

 

[rl.bhat]

At any instant, acceleration is given as
g = dv/dt= dv/dh*dh/dt = v*dv/dh = g_o - hg'
Now find the integration to find v in terms of h.
Now g = g_o - hg' = g_o + ( h_o - 1/2*gt²)g'
Collect the terms containing g and write g = ...
Then write g = dv/dt and find the integration to find v in terms of t.