There is a train A. On the roof of A is another frictionless train B of mass Mb. A mass Mc hangs off the front of A and is attached to the front of B by rope and frictionless pulley. How fast should A accelerate so that B wont fall off the roof of A.

You must carefully draw a diagram. The key to this question is to find from what perspective you should view it. The easiest way is to see that B wants to accelerate due to the weight of C. This is unaffected by the motion of A. We simply need to accelerate A at the same rate that B would accelerate so there will be no relative motion and so B wont fall off. We have the acceleration of b = (Mc*g)/(Mb+Mc) So that is what the acceleration of A should be.