Chapter 1.3, Problem 22E

In Problem 21, the mass m(t) is the sum of three different masses: m(t) = m_p + m_v + m_f(t), where m_p is the constant mass of the payload, m_v is the constant mass of the vehicle, and m_f(t) is the variable amount of fuel.

1. (a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass m_f(t) of the fuel changes.
2. (b) If the rocket consumes its fuel at a constant rate λ, find m(t). Then rewrite the differential equation in Problem 21 in terms of λ and the initial total mass m(0) = m₀.
3. (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.