A rocket with a total mass of 6000 kg has half of its mass in fuel. It can expel the fuel with a velocity of 2000 m/s relative to the rocket. The rocket is to land on a planet that has a constant acceleration due to gravity of 3 m/s^2. When the rocket is 40,000 m above the planet, it is descending at 400 m/s. The rocket fires its engine to slow its descent and regulates the fuel flow to give it a constant upward acceleration of a. (i) Use simple kinematics to show a = 2 m/s^2. (ii) Start from the impulse momentum theorem to derive: -mgdt (m + dm)(V + dV) + (-dm)(V - Ve) - mV (look at your notes) (ii) Show that the mass of the rocket+fuel as a function of time is given by m = 6000exp(-5t/2000) (iii) Is there enough fuel to complete this landing? If so, how much fuel does the rocket have on landing? If not, what is its crash speed?