In Problem 29, suppose m(t) = mp + mν + mf (t), where mp is constant mass of the payload...

In Problem 29, suppose m(t) = m_p + m_ν + m_f (t), where m_p is constant mass of the payload, m_ν is the constant mass of the vehicle, and m_f (t) is the variable amount of fuel.

(a) Show that the rate at which the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes.

(b) If the rocket consumes its fuel at a constant rate λ, find m(t). Then rewrite the differential equation in Problem 29 in terms of λ and the initial total mass m(0) = m₀.

(c) Under the assumption in part (b), show that the burnout time t_b > 0 of the rocket, or the time at which all the fuel is consumed, is t_b = m_f (0)/λ, where m_f (0) is the initial mass of the fuel.

Reference:

Problem 29:

Consider a small single-stage rocket launched vertically. Let m(t) denote the total mass of the rocket at time t (which is the sum of three masses: the constant mass of the payload, the constant mass of the vehicle, and the variable mount of fuel). If it is assumed that the positive direction is upward, air resistance is proportional to the nstantaneous velocity ν of the rocket, and R is the upward thrust or force generated by the propulsion system, then find a mathematical model for the velocity ν(t) of the rocket. [Hint: See (14) in Section 1.3 and (18) in Exercises 1.3.]