Hi,

You problem is long to read and I do not have much time. But for question a) and b), you have the equation of d2h/d2t, something in 1/t. The second antiderivative is tln(t)-ln(t). So you can easily find h(t) (will be written using this function). After you have h(0)=0 and I think you meant dh(0)/dt=0 also. So you can find h(t) and also dh/t at any time. Now your condition for end of bust is that bt=moX, that gives you the time and you can find the height and the velocity at that moment. If you have problem with question c) (I did not even read it!) let us know, I will try to have a look later ( I used to like these rockets problem, and this one is actually not hard as the equation is already given, what is more interesting is to find the equation of acceleration!)