Consider a rocket with total initial mass m exactly half of m is fuel, which is expelled from the rocket at a constant rate k (mass per unit time) and with speed with respect to the rocket. there is a uniform gravitational field g downwards, and the motion of function of time be r(t). the the rocket is entirely vertical. let the height of the rocket as a the ground (at rocket is at rest on 0) and begins expelling fuel at time t = 0. (a) 16 points find (b) 6 points find the condition on differential equation satisfied by the height of the rocket a m, k, u, g for the rocket to lift off immediately at t = 0, and the condition for the rocket to lift off at all (at any t) (c) 18 points assuming that the rocket takes off immediately at t 0, find expressions for the height and speed of the rocket at the moment when it runs out of fuel. write your algebraic expressions in terms of only the quantities m, k, u, g. answers as the following indefinite integral may or may not be useful n(1 az)dz (z - 1/a) in(1 az) 2.