Chapter 13.4, Problem 46E

A rocket burning its onboard fuel while moving through space has velocity v(t) and mass m(t) at time t. If the exhaust gases escape with velocity v_e relative to the rocket, it can be deduced from Newton’s Second Law of Motion that

$m\frac{d\text{v}}{dt}=\frac{dm}{dt}{\text{v}}_e$

(a) Show that $\text{v}(t)=\text{v}(0)-\ln \frac{m(0)}{m(t)}{\text{v}}_e$.

(b) For the rocket to accelerate in a straight line from rest to twice the speed of its own exhaust gases, what fraction of its initial mass would the rocket have to bum as fuel?