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 ve relative to the rocket, it can be deduced from Newton’s Second Law of Motion that

m dv/dt = dm/dt v_e
(a) Show that v(t) = v(0) - ln m(0)/m(t) 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 burn as fuel?