Acylindrical rocket of diameter 2.r, mass m, and containing fuel of mass m, is moving through empty space at velocity vu. at some point the rocket enters a unifom cloud of interstellar particles with volume density n (e g. particles/m), each particle having the mass m < ms and initially at rest. to compensate for the dissipative force of the particles colliding with the rocket, the rocket engines emit fuel at the rate y with dm, /dr =-7 at a constant velocity u with respect to the rocket. ignore gravita- tional interaction between the rocket and the cloud particles. 1. assuming that the dissipative drag force from the cloud particles is -av, where a is a constant, derive the differential equation for the velocity vi) of the rocket through the cloud as it is firing its engines 2. what must the rocket thrust yu be to maintain the constant velocity u7 3, if the rocket suddenly runs out of fuel, how would its velocity vn) decrease with time as the rocket moves through the cloud. 4. bonus: assuming that each cloud particle bounces off the rocket elastically, and collisions occur very frequently, prove that the drag force is indeed proportional to v and estimate the constant a, if the front nose-cone of the rocket has the opening angle of 90 n particles/m tg