A weather rocket of mass m is launched vertically up from a horizontal platform. The rocket engine burned for a short time T with a time dependent thrust F(t)=bt^2, where b is a constant and t is the time. Assuming air friction is negligible,
(a) What is the height h of the rocket when the engine is turned off at time T?
(b) What is the maximum height Hm the rocket can reach? Express h and Hm in terms of b, T, g, and m.