A small rocket, of the sort that the high school boys built in the 1998 movie ''October Sky'', has a mass, M. In addition, at firing, it carries a mass, mf , of fuel, which is burned at a rate dm/dt , where m(t) is the total mass of the rocket. The combustion products from the burning fuel are expelled from a nozzle at the rear of the rocket at a constant speed, u, with respect to the rocket. Assume that the rocket goes essentially straight up, that its maximum altitude is small enough that the acceleration of gravity can be considered to be constant, and that one can ignore air resistance.
a) From Newton's Second Law, write a first- order differential equation for the vertical velocity, v(t), of the rocket.